V235.XGP[TEX,DEK]2 - www.SailDart.org

perm filename V235.XGP[TEX,DEK]2 blob sn#510441 filedate 1980-05-16 generic text, type T, neo UTF8
/NOWRAPAROUND/LMAR=50/TMAR=50/RMAR=1700/BMAR=1/PMAR=0/XLINE=0/FONT#0=NGR13/USETI=000000275*TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX*

␈β	[␈↓ α4␈ε"SECTION␈α3.5␈αof␈αTHE␈αAR␈α⎇T␈αOF␈αCOMPUTER␈αPR␈α␈OGRAMMING
␈β
ε␈↓ β&␈ε6⎇␈ε"␈α1980␈αAddison↑Wesley␈αPublishing␈αCompan␈α␈y,␈αInc.
␈β⊃H␈↓ ε2␈ε$0
␈β∪(

␈β↓U␈↓ ↓H␈ε"142␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα$␈↓ ↓H␈ε=3␈α␈.5.␈α∩WHA␈α⎇T␈α
IS␈α
A␈α
RANDOM␈α
SE␈α␈QUENCE?
␈βαf␈↓ ↓H␈ε2A.␈αλIn␈α␈troductory␈α	remarks.␈ε"␈α∞We␈↓ ¬∃␈ε"ha␈α␈v␈α␈e␈α	seen␈α	in␈α	this␈αλchapter␈α	h␈α↓o␈α␈w␈α	to␈α	generate␈αλsequences
␈ββ=␈↓ ¬π␈ε6h␈↓ ¬∪␈ε(U␈↓ ¬?␈ε6i␈ε"␈α
=␈↓ εβ␈ε(U␈↓ ε+␈ε",␈↓ εC␈ε(U␈↓ εk␈ε",␈↓ πβ␈ε(U␈↓ π+␈ε",␈↓ πC␈ε".␈αε.␈αε.␈↓ α␈ε"(1)
␈ββJ␈↓ ¬,␈ε+n␈↓ ε≤␈ε%0␈↓ ε[␈ε%1␈↓ π≠␈ε%2
␈β∧∀␈↓ ↓H␈ε"of␈α
real␈α
n␈α␈um␈α␈bers␈α∞in␈α
the␈α
range␈α∞0␈ε6␈α∀␈↓ ¬h␈ε(U␈↓ ε ␈ε"<␈α1,␈α∞and␈α
w␈α␈e␈α
ha␈α␈v␈α␈e␈α∞called␈α
them␈α
\random"
␈β∧!␈↓ ε↓␈ε+n
␈β∧@␈↓ ↓H␈ε"sequences␈α∞ev␈α␈en␈α
th␈α↓ough␈α∞they␈α∞are␈α∞completely␈α∞deterministic␈α∞in␈α∞character.␈α⊗T␈α⎇o␈α
jus-
␈β∧k␈↓ ↓H␈ε"tify␈αthis␈αtermin␈α↓ology,␈α
w␈α␈e␈αclaimed␈αthat␈αthe␈α
n␈α␈um␈α␈bers␈α\beha␈α␈v␈α␈e␈αas␈α
if␈αthey␈αare␈αtruly
␈β¬⊗␈↓ ↓H␈ε"random."␈α⊗Such␈α∞a␈α∞statemen␈α␈t␈α∞ma␈α␈y␈α∞be␈α∞satisfactory␈α∞for␈α∞practical␈α∞purposes␈α∞(at␈α∞the
␈β¬A␈↓ ↓H␈ε"presen␈α␈t␈α⊂time),␈α⊃but␈α⊂it␈α⊂sidesteps␈α⊂a␈α⊂v␈α␈ery␈α⊂importan␈α␈t␈α⊂philosophical␈α⊂and␈α⊂theoretical
␈β¬l␈↓ ↓H␈ε"question:␈α∪Precisely␈α⊂what␈α⊂do␈α⊂w␈α␈e␈α⊂mean␈α⊂by␈α⊂\random␈α⊂beha␈α␈vior"?␈α≤A␈α∂quan␈α␈titativ␈α␈e
␈βε_␈↓ ↓H␈ε"de|nition␈α⊂is␈α∂needed.␈α≤It␈α⊂is␈α⊂undesirable␈α⊂to␈α⊂talk␈α⊂about␈α⊂concepts␈α⊂that␈α⊂w␈α␈e␈α⊂do␈α∂n␈α↓ot
␈βεC␈↓ ↓H␈ε"really␈αunderstand,␈αespecially␈αsince␈αman␈α␈y␈αapparen␈α␈tly␈αparado␈α␈xical␈αstatemen␈α␈ts␈αcan
␈βεn␈↓ ↓H␈ε"be␈αmade␈αabout␈αrandom␈αn␈α␈um␈α␈bers.
␈βπ→␈↓ α␈ε"The␈α⊃mathematical␈α⊃theory␈α⊃of␈α⊃probabilit␈α␈y␈α⊃and␈α⊃statistics␈α⊃carefully␈α⊃sidesteps
␈βπE␈↓ ↓H␈ε"the␈α
question;␈αit␈αrefrains␈αfrom␈αmaking␈α
absolute␈αstatemen␈α␈ts,␈αand␈αinstead␈α
expresses
␈βπp␈↓ ↓H␈ε"ev␈α␈erything␈αin␈αterms␈αof␈αh␈α↓o␈α␈w␈αm␈α␈uch␈ε/␈αprobabilit␈α␈y␈ε"␈αis␈αto␈αbe␈αattached␈αto␈αstatemen␈α␈ts␈αin-
␈βλ≠␈↓ ↓H␈ε"v␈α␈olving␈α
random␈α
sequences␈α
of␈α
ev␈α␈en␈α␈ts.␈α∂The␈α
axioms␈α
of␈α
probabilit␈α␈y␈α
theory␈α
are␈α
set␈α
up
␈βλF␈↓ ↓H␈ε"so␈α
that␈α
abstract␈α
probabilities␈α
can␈α
be␈α
computed␈α
readily,␈α
but␈α
n␈α↓othing␈α
is␈α
said␈α
about
␈βλq␈↓ ↓H␈ε"what␈α	probabilit␈α␈y␈α	really␈α
signi|es,␈α	or␈α
h␈α↓o␈α␈w␈α	this␈α	concept␈α
can␈α	be␈α	applied␈α	meaningfully
␈β	≥␈↓ ↓H␈ε"to␈α
the␈α
actual␈α
w␈α␈orld.␈α∪In␈α
the␈α
bo␈α↓ok␈ε/␈α
Probabilit␈α␈y,␈α
Statistics,␈α
and␈α
T␈α⎇ruth␈ε"␈α
(New␈α
Y␈α⎇ork:
␈β	H␈↓ ↓H␈ε"Macmillan,␈α∞1957),␈α∂R.␈↓ ∧%␈ε"v␈α␈on␈α∂Mises␈α∞discusses␈α∞this␈α∞situation␈α∞in␈α∂detail,␈α∞and␈α∞presen␈α␈ts
␈β	s␈↓ ↓H␈ε"the␈α
view␈α∞that␈α∞a␈α
proper␈α∞de|nition␈α∞of␈α
probabilit␈α␈y␈α∞depends␈α
on␈α∞obtaining␈α∞a␈α
proper
␈β
≡␈↓ ↓H␈ε"de|nition␈αof␈αa␈αrandom␈αsequence.
␈β
J␈↓ α␈ε"Let␈αus␈α
paraphrase␈αhere␈αsome␈αstatemen␈α␈ts␈α
made␈αby␈αt␈α␈w␈α␈o␈α
of␈αthe␈αman␈α␈y␈αauth␈α↓ors
␈β
u␈↓ ↓H␈ε"wh␈α↓o␈αha␈α␈v␈α␈e␈αcommen␈α␈ted␈αon␈αthe␈αsubject.
␈βL␈↓ α␈ε/D.␈αH.␈↓ αl␈ε/Lehmer␈ε"␈α(1951):␈α\A␈αrandom␈αsequence␈αis␈αa␈αv␈α⎇ague␈αn␈α↓otion␈αem␈α␈bodying␈αthe
␈βx␈↓ α␈ε"idea␈α∂of␈α∂a␈α∂sequence␈α∂in␈α∞which␈α∂each␈α∂term␈α∂is␈α∂unpredictable␈α∂to␈α∂the␈α∞uninitiated
␈β#␈↓ α␈ε"and␈α	wh␈α↓ose␈α	digits␈α
pass␈α	a␈α	certain␈α	n␈α␈um␈α␈ber␈α	of␈α	tests,␈α
traditional␈α	with␈α	statisticians
␈βN␈↓ α␈ε"and␈αdepending␈αsomewhat␈αon␈αthe␈αuses␈αto␈αwhich␈αthe␈αsequence␈αis␈αto␈αbe␈αput."
␈β
⊃␈↓ α␈ε/J.␈αN.␈↓ αg␈ε/F␈α⎇ranklin␈ε"␈α(1962):␈α
\The␈α
sequence␈α(1)␈α
is␈αrandom␈α
if␈αit␈α
has␈αev␈α␈ery␈αpropert␈α␈y
␈β
<␈↓ α␈ε"that␈α∂is␈α∂shared␈α∞by␈α∂all␈α∂in|nite␈α∂sequences␈α∞of␈α∂independen␈α␈t␈α∂samples␈α∂of␈α∞random
␈β
g␈↓ α␈ε"v␈α⎇ariables␈αfrom␈αthe␈αuniform␈αdistribution."
␈β∞>␈↓ α␈ε"F␈α⎇ranklin's␈α∩statemen␈α␈t␈α∩essen␈α␈tially␈α∩generalizes␈α⊃Lehmer's␈α∩to␈α∩sa␈α␈y␈α∩that␈α∩the␈α⊃se-
␈β∞j␈↓ ↓H␈ε"quence␈αm␈α␈ust␈αsatisfy␈ε/␈αall␈ε"␈αstatistical␈αtests.␈α⊂His␈αde|nition␈αis␈αn␈α↓ot␈αcompletely␈αprecise,
␈β∂∃␈↓ ↓H␈ε"and␈α
w␈α␈e␈α
will␈α
see␈α
later␈α
that␈α
a␈αreasonable␈α
in␈α␈terpretation␈α
of␈α
his␈α
statemen␈α␈t␈α
leads␈α
us␈α
to
␈β∂@␈↓ ↓H␈ε"conclude␈αthat␈αthere␈αis␈αn␈α↓o␈αsuch␈αthing␈αas␈αa␈αrandom␈αsequence!␈α∂So␈αlet␈αus␈αbegin␈αwith
␈β∂k␈↓ ↓H␈ε"Lehmer's␈α∂less␈α∂restrictiv␈α␈e␈α∂statemen␈α␈t␈α∂and␈α∂attempt␈α∂to␈α∂mak␈α␈e␈ε/␈α∂it␈ε"␈α∂precise.␈α→What␈α∂w␈α␈e
␈β⊂⊗␈↓ ↓H␈ε"really␈αw␈α␈an␈α␈t␈αis␈αa␈αrelativ␈α␈ely␈αsh␈α↓ort␈αlist␈αof␈αmathematical␈αproperties,␈αeach␈αof␈αwhich␈αis
␈β⊂B␈↓ ↓H␈ε"satis|ed␈α
by␈αour␈α
in␈α␈tuitiv␈α␈e␈α
n␈α↓otion␈αof␈α
a␈αrandom␈α
sequence;␈αfurtherm␈α↓ore,␈αthe␈αlist␈α
is␈α
to
␈β⊂m␈↓ ↓H␈ε"be␈αcomplete␈αen␈α↓ough␈αso␈αthat␈αw␈α␈e␈αare␈αwilling␈αto␈αagree␈αthat␈ε/␈αan␈α␈y␈ε"␈αsequence␈αsatisfying
␈β⊃_␈↓ ↓H␈ε"these␈αproperties␈α
is␈α\random."␈α∩In␈αthis␈α
section,␈αw␈α␈e␈α
will␈αdev␈α␈elop␈α
what␈αseems␈α
to␈αbe
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"143
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα$␈↓ ↓H␈ε"an␈α
adequate␈α
de|nition␈αof␈α
randomness␈α
according␈αto␈α
these␈α
criteria,␈αalth␈α↓ough␈α
man␈α␈y
␈βαO␈↓ ↓H␈ε"in␈α␈teresting␈αquestions␈αremain␈αto␈αbe␈αansw␈α␈ered.
␈βαz␈↓ α␈ε"Let␈ε(␈α∞u␈ε"␈α
and␈↓ β:␈ε(v␈↓ β[␈ε"be␈α∞real␈α∞n␈α␈um␈α␈bers,␈α∞0␈ε6␈α
∀␈ε(␈α
u␈ε"␈α
<␈↓ π∀␈ε(v␈↓ π4␈ε6∀␈ε"␈α
1.␈α∃If␈↓ λ<␈ε(U␈↓ λg␈ε"is␈α∞a␈α∞random␈α
v␈α⎇ariable
␈ββ&␈↓ ↓H␈ε"that␈αis␈αuniformly␈α
distributed␈αbet␈α␈w␈α␈een␈α
0␈αand␈α1,␈α
the␈αprobabilit␈α␈y␈α
that␈ε(␈αu␈ε6␈α
∀␈↓ 
C␈ε(U␈↓ 
k␈ε"<␈↓ ~␈ε(v
␈ββL␈↓ λ␈ε%1␈↓ 	1␈ε%2␈↓ 	x␈ε%1
␈ββQ␈↓ ↓H␈ε"is␈αequal␈αto␈↓ αv␈ε(v␈↓ β⊃␈ε6␈␈ε(␈αλu␈ε"␈α␈.␈α⊂F␈α⎇or␈αexample,␈αthe␈αprobabilit␈α␈y␈αthat␈↓ λ*␈ε6∀␈↓ λX␈ε(U␈↓ λ␈␈ε"<␈↓ 	P␈ε"is␈↓ 
␈ε".␈α⊂Ho␈α␈w␈αcan
␈ββb␈↓ λ␈ε%3␈↓ 	1␈ε%3␈↓ 	x␈ε%3
␈ββe␈↓ λ␈∧βeλα⊂␈↓ 	1␈∧βe	1α⊂␈↓ 	x␈∧βe	xα⊂
␈ββ|␈↓ ↓H␈ε"w␈α␈e␈αtranslate␈αthis␈αpropert␈α␈y␈αof␈αthe␈αsingle␈αn␈α␈um␈α␈ber␈↓ π6␈ε(U␈↓ π←␈ε"in␈α␈to␈αa␈αpropert␈α␈y␈αof␈αthe␈αin|nite
␈β∧'␈↓ ↓H␈ε"sequence␈↓ α]␈ε(U␈↓ β¬␈ε",␈↓ β≥␈ε(U␈↓ βE␈ε",␈↓ β]␈ε(U␈↓ ∧¬␈ε",␈↓ ∧≤␈ε".␈αε.␈αε.␈↓ ∧L␈ε"?␈α∃The␈α
obvious␈α∞answ␈α␈er␈α
is␈α∞to␈α
coun␈α␈t␈α∞h␈α↓o␈α␈w␈α
man␈α␈y␈α∞times␈↓ ↓␈ε(U
␈β∧4␈↓ αv␈ε%0␈↓ β6␈ε%1␈↓ βu␈ε%2␈↓ ~␈ε+n
␈β∧R␈↓ ↓H␈ε"lies␈αbet␈α␈w␈α␈een␈ε(␈αu␈ε"␈αand␈↓ βt␈ε(v␈↓ ∧π␈ε",␈αand␈αthe␈αa␈α␈v␈α␈erage␈αn␈α␈um␈α␈ber␈αof␈αtimes␈↓ λ+␈ε"sh␈α↓ould␈αequal␈↓ 	{␈ε(v␈↓ 
∃␈ε6␈␈ε(␈απu␈ε".␈α∂Our
␈β∧}␈↓ ↓H␈ε"in␈α␈tuitiv␈α␈e␈αidea␈αof␈αprobabilit␈α␈y␈αis␈αbased␈αin␈αthis␈αw␈α␈a␈α␈y␈αon␈αthe␈αfrequency␈αof␈αoccurrence.
␈β¬)␈↓ α␈ε"More␈α⊂precisely,␈α⊂let␈↓ ∧E␈ε(↔␈↓ ∧Z␈ε"(␈ε(n␈ε")␈α∂be␈α⊂the␈α⊂n␈α␈um␈α␈ber␈α∂of␈α⊂v␈α⎇alues␈α⊂of␈↓ λ[␈ε(j␈↓ λn␈ε",␈α⊃0␈ε6␈α⊂∀␈↓ 	←␈ε(j␈↓ 
α␈ε"<␈ε(␈α⊂n␈ε",␈α⊂such
␈β¬T␈↓ ↓H␈ε"that␈ε(␈α
u␈ε6␈α∀␈↓ αh␈ε(U␈↓ β≥␈ε"<␈↓ βM␈ε(v␈↓ β`␈ε";␈α∞w␈α␈e␈α∞w␈α␈an␈α␈t␈α
the␈α
ratio␈↓ ε∨␈ε(↔␈↓ ε4␈ε"(␈ε(n␈ε")/␈ε(n␈ε"␈α
to␈α
approach␈α∞the␈α
v␈α⎇alue␈↓ 	}␈ε(v␈↓ 
~␈ε6␈␈ε(␈α	u␈ε"␈α
as␈ε(␈α∞n
␈β¬`␈↓ β↓␈ε+j
␈β¬␈␈↓ ↓H␈ε"approaches␈αin|nit␈α␈y:
␈βε*␈↓ ¬~␈ε"lim␈↓ ¬`␈ε(↔␈↓ ¬u␈ε"(␈ε(n␈ε")/␈ε(n␈ε"␈α	=␈↓ πα␈ε(v␈↓ π≥␈ε6␈␈ε(␈αλu␈ε".␈↓ α␈ε"(2)
␈βεK␈↓ ¬␈ε+n␈ε9!␈α␈1
␈βεz␈↓ ↓H␈ε"If␈α⊃this␈α⊂condition␈α⊃h␈α↓olds␈α⊃for␈α⊃all␈α⊃ch␈α↓oices␈α⊃of␈ε(␈α⊃u␈ε"␈α⊂and␈↓ πU␈ε(v␈↓ πh␈ε",␈α∩the␈α⊃sequence␈α⊃is␈α⊃said␈α⊃to␈α⊂be
␈βπ&␈↓ ↓H␈ε/equidistributed.
␈βπQ␈↓ α␈ε"Let␈↓ αO␈ε(S␈↓ αh␈ε"(␈ε(n␈ε")␈α
be␈αa␈α
statemen␈α␈t␈α
about␈α
the␈α
in␈α␈teger␈ε(␈αn␈ε"␈α
and␈α
the␈α
sequence␈↓ 	t␈ε(U␈↓ 
≤␈ε",␈↓ 
3␈ε(U␈↓ 
[␈ε",␈↓ 
r␈ε".␈αε.␈αε.␈↓ "␈ε";
␈βπ]␈↓ 

␈ε%1␈↓ 
L␈ε%2
␈βπ|␈↓ ↓H␈ε"for␈α
example,␈↓ β∃␈ε(S␈↓ β.␈ε"(␈ε(n␈ε")␈α
migh␈α␈t␈αbe␈α
the␈αstatemen␈α␈t␈↓ ε\␈ε"considered␈αabo␈α␈v␈α␈e,␈αnamely␈α
\␈ε(u␈ε6␈α
∀␈↓ 
S␈ε(U␈↓ λ␈ε"<
␈βλλ␈↓ 
l␈ε+n
␈βλ'␈↓ ↓H␈ε(v␈↓ ↓[␈ε"."␈α∀We␈α
can␈α
generalize␈α
the␈α∞idea␈α
used␈α
in␈α∞the␈α
preceding␈α
paragraph␈α∞to␈α
de|ne␈α
\the
␈βλR␈↓ ↓H␈ε"probabilit␈α␈y␈αthat␈↓ βN␈ε(S␈↓ βg␈ε"(␈ε(n␈ε"␈α␈)␈αis␈αtrue"␈αwith␈αrespect␈αto␈αa␈αparticular␈αin|nite␈αsequence:␈αLet
␈βλ}␈↓ ↓H␈ε(↔␈↓ ↓]␈ε"(␈ε(n␈ε")␈αbe␈αthe␈αn␈α␈um␈α␈ber␈αof␈αv␈α⎇alues␈αof␈↓ ¬E␈ε(j␈↓ ¬X␈ε",␈α0␈ε6␈α
∀␈↓ ε8␈ε(j␈↓ εU␈ε"<␈ε(␈α
n␈ε"␈α␈,␈αsuch␈αthat␈↓ λN␈ε(S␈↓ λg␈ε"(␈↓ λs␈ε(j␈↓ 	ε␈ε")␈αis␈αtrue.
␈β	≥␈↓ ∧g␈ε↓␈␈↓ ¬<␈ε↓↓
␈β	<␈↓ ↓H␈ε2De|nition␈α⊂A.␈ε/␈α"We␈α⊂sa␈α␈y␈↓ ∧A␈ε"P␈↓ ∧Y␈ε"r␈↓ ∧u␈ε(S␈↓ ¬∞␈ε"(␈ε(n␈ε")␈↓ ¬[␈ε"=␈ε(␈α∩∃␈ε/,␈α∩if␈↓ εo␈ε"lim␈↓ πt␈ε(↔␈↓ λ
␈ε"(␈ε(n␈ε"␈α␈)/␈ε(n␈ε"␈α∩=␈ε(␈α⊃∃␈ε/.␈ε"␈α≡(Read,␈↓ 
\␈ε"\The
␈β	H␈↓ π!␈ε+n␈ε9␈α␈!1
␈β	g␈↓ ↓H␈ε"probabilit␈α␈y␈α
that␈↓ βJ␈ε(S␈↓ βc␈ε"(␈ε(n␈ε")␈α
is␈αtrue␈α
equals␈ε(␈α
∃␈ε",␈αif␈α
the␈α
limit␈α
as␈ε(␈αn␈ε"␈α
tends␈α
to␈α
in|nit␈α␈y␈αof␈↓ 
B␈ε(↔␈↓ 
W␈ε"(␈ε(n␈ε")/␈ε(n
␈β
∩␈↓ ↓H␈ε"equals␈ε(␈α∃␈ε".")
␈β
P␈↓ α␈ε"In␈α
terms␈αof␈α
this␈α
n␈α↓otation,␈α
the␈αsequence␈↓ ε}␈ε(U␈↓ π&␈ε",␈↓ π=␈ε(U␈↓ πe␈ε",␈↓ π|␈ε".␈αε.␈αε.␈↓ λ2␈ε"is␈α
equidistributed␈α
if␈αand
␈β
]␈↓ π↔␈ε%0␈↓ πV␈ε%1
␈β
{␈↓ ↓H␈ε"only␈α
if␈↓ α6␈ε"P␈↓ αN␈ε"r␈↓ α]␈ε"(␈ε(u␈ε6␈α	∀␈↓ β5␈ε(U␈↓ βk␈ε"<␈↓ ∧→␈ε(v␈↓ ∧,␈ε")␈α
=␈↓ ∧p␈ε(v␈↓ ¬λ␈ε6␈␈ε(␈αεu␈ε"␈α␈,␈αfor␈αall␈αreal␈αn␈α␈um␈α␈bers␈ε(␈αu␈ε"␈α␈,␈↓ λ@␈ε(v␈↓ λ↑␈ε"with␈α0␈ε6␈α
∀␈ε(␈α
u␈ε"␈α
<␈↓ 
F␈ε(v␈↓ 
b␈ε6∀␈ε"␈α
1.
␈βλ␈↓ βN␈ε+n
␈β'␈↓ α␈ε"A␈α
sequence␈αma␈α␈y␈α
be␈α
equidistributed␈αwith␈α↓out␈α
being␈α
random.␈α∩F␈α⎇or␈α
example,␈αif
␈βR␈↓ ↓H␈ε(U␈↓ ↓p␈ε",␈↓ αε␈ε(U␈↓ α.␈ε",␈↓ αE␈ε".␈αε.␈αε.␈↓ α|␈ε"and␈↓ βB␈ε(V␈↓ βg␈ε",␈↓ β}␈ε(V␈↓ ∧#␈ε",␈↓ ∧:␈ε".␈αε.␈αε.␈↓ ∧p␈ε"are␈α
equidistributed␈α
sequences,␈αit␈α
is␈α
n␈α↓ot␈αhard␈α
to␈αsh␈α↓o␈α␈w
␈β↑␈↓ ↓`␈ε%0␈↓ α∨␈ε%1␈↓ βX␈ε%0␈↓ ∧∀␈ε%1
␈β⎇␈↓ ↓H␈ε"that␈αthe␈αsequence
␈βI␈↓ ¬n␈ε%1␈↓ εF␈ε%1␈↓ π¬␈ε%1␈↓ πY␈ε%1␈↓ λ0␈ε%1␈↓ λo␈ε%1
␈βM␈↓ β␈ε(W␈↓ β9␈ε",␈↓ βI␈ε(W␈↓ βw␈ε",␈↓ ∧π␈ε(W␈↓ ∧4␈ε",␈↓ ∧D␈ε(W␈↓ ∧r␈ε",␈↓ ¬α␈ε".␈αε.␈αε.␈↓ ¬<␈ε"=␈↓ εα␈ε(U␈↓ ε*␈ε",␈↓ ε[␈ε"+␈↓ π_␈ε(V␈↓ π=␈ε",␈↓ πl␈ε(U␈↓ λ∀␈ε",␈↓ λE␈ε"+␈↓ 	α␈ε(V␈↓ 	'␈ε",␈↓ 	?␈ε".␈αε.␈αε.␈↓ α␈ε"(3)
␈βZ␈↓ β)␈ε%0␈↓ βg␈ε%1␈↓ ∧%␈ε%2␈↓ ∧c␈ε%3␈↓ ε~␈ε%0␈↓ π.␈ε%0␈↓ λ¬␈ε%1␈↓ 	_␈ε%1
␈β↑␈↓ ¬n␈ε%2␈↓ εF␈ε%2␈↓ π¬␈ε%2␈↓ πY␈ε%2␈↓ λ0␈ε%2␈↓ λo␈ε%2
␈βb␈↓ ¬n␈∧b¬nα⊂␈↓ εF␈∧bεFα⊂␈↓ π¬␈∧bπ¬α⊂␈↓ πY␈∧bπYα⊂␈↓ λ0␈∧bλ0α⊂␈↓ λo␈∧bλoα⊂
␈β
→␈↓ π0␈ε%1␈↓ λ
␈ε%1
␈β
≡␈↓ ↓H␈ε"is␈α∂also␈α⊂equidistributed,␈α⊂since␈α⊂the␈α∂subsequence␈↓ πC␈ε(U␈↓ πk␈ε",␈↓ λ≥␈ε(U␈↓ λE␈ε",␈↓ λ←␈ε".␈αε.␈αε.␈↓ 	→␈ε"is␈α∂equidistributed
␈β
*␈↓ π\␈ε%0␈↓ λ6␈ε%1
␈β
/␈↓ π0␈ε%2␈↓ λ
␈ε%2
␈β
2␈↓ π0␈∧
2π0α⊂␈↓ λ
␈∧
2λ
α⊂
␈β
D␈↓ βA␈ε%1␈↓ π∀␈ε%1␈↓ πc␈ε%1␈↓ λ8␈ε%1␈↓ 	π␈ε%1
␈β
I␈↓ ↓H␈ε"bet␈α␈w␈α␈een␈α∞0␈α∂and␈↓ βU␈ε",␈α∂while␈α∂the␈α∞alternate␈α∂terms␈↓ π1␈ε"+␈↓ πv␈ε(V␈↓ λ≠␈ε",␈↓ λU␈ε"+␈↓ 	~␈ε(V␈↓ 	?␈ε",␈↓ 	Y␈ε".␈αε.␈αε.␈↓ 
	␈ε",␈α∂are␈α∞equi-
␈β
U␈↓ λ␈ε%0␈↓ 	0␈ε%1
␈β
Z␈↓ βA␈ε%2␈↓ π∀␈ε%2␈↓ πc␈ε%2␈↓ λ8␈ε%2␈↓ 	π␈ε%2
␈β
]␈↓ βA␈∧
]βAα⊂␈↓ π∀␈∧
]π∀α⊂␈↓ πc␈∧
]πcα⊂␈↓ λ8␈∧
]λ8α⊂␈↓ 	π␈∧
]	πα⊂
␈β
p␈↓ ∧⊗␈ε%1␈↓ 
q␈ε%1
␈β
t␈↓ ↓H␈ε"distributed␈α⊂bet␈α␈w␈α␈een␈↓ ∧9␈ε"and␈α⊃1.␈α≤In␈α⊃the␈α⊂sequence␈α⊂of␈↓ πu␈ε(W␈↓ λ→␈ε"'s,␈α⊃a␈α⊂v␈α⎇alue␈α⊃less␈α⊂than␈↓ ∀␈ε"is
␈β∞¬␈↓ ∧⊗␈ε%2␈↓ 
q␈ε%2
␈β∞λ␈↓ ∧⊗␈∧∞λ∧⊗α⊂␈↓ 
q␈∧∞λ
qα⊂
␈β∞≠␈↓ πt␈ε%1
␈β∞∨␈↓ ↓H␈ε"alw␈α␈a␈α␈ys␈αfollo␈α␈w␈α␈ed␈αby␈αa␈αv␈α⎇alue␈αgreater␈αthan␈αor␈αequal␈αto␈↓ λπ␈ε",␈αand␈αcon␈α␈v␈α␈ersely;␈αhence␈αthe
␈β∞0␈↓ πt␈ε%2
␈β∞3␈↓ πt␈∧∞3πtα⊂
␈β∞K␈↓ ↓H␈ε"sequence␈αis␈α
n␈α↓ot␈αrandom␈α
by␈αan␈α␈y␈α
reasonable␈αde|nition.␈α∩A␈αstronger␈α
propert␈α␈y␈αthan
␈β∞v␈↓ ↓H␈ε"equidistribution␈αis␈αneeded.
␈β∂!␈↓ α␈ε"A␈α⊃natural␈α⊃generalization␈α⊃of␈α⊃the␈α⊃equidistribution␈α⊂propert␈α␈y,␈α∪which␈α⊂rem␈α↓o␈α␈v␈α␈es
␈β∂L␈↓ ↓H␈ε"the␈αobjection␈αstated␈αin␈αthe␈αpreceding␈αparagraph,␈αis␈αto␈αconsider␈αadjacen␈α␈t␈αpairs␈αof
␈β∂w␈↓ ↓H␈ε"n␈α␈um␈α␈bers␈αof␈αour␈αsequence.␈α⊂We␈αcan␈αrequire␈αthe␈αsequence␈αto␈αsatisfy␈αthe␈αcondition
␈β⊂H␈↓ α7␈ε"P␈↓ αO␈ε"r␈↓ α]␈ε"(␈↓ αi␈ε(u␈↓ β↔␈ε6∀␈↓ βE␈ε(U␈↓ βz␈ε"<␈↓ ∧(␈ε(v␈↓ ∧m␈ε"and␈↓ ¬K␈ε(u␈↓ ¬y␈ε6∀␈↓ ε'␈ε(U␈↓ π	␈ε"<␈↓ π7␈ε(v␈↓ πX␈ε")␈α
=␈α
(␈↓ λ(␈ε(v␈↓ λQ␈ε6␈␈↓ λ⎇␈ε(u␈↓ 	!␈ε")(␈↓ 	9␈ε(v␈↓ 	a␈ε6␈␈↓ 

␈ε(u␈↓ 
1␈ε")␈↓ α␈ε"(4)
␈β⊂T␈↓ α}␈ε%1␈↓ β↑␈ε+n␈↓ ∧:␈ε%1␈↓ ¬`␈ε%2␈↓ ε@␈ε+n␈ε%␈α␈+1␈↓ πI␈ε%2␈↓ λ9␈ε%1␈↓ 	⊃␈ε%1␈↓ 	J␈ε%2␈↓ 
"␈ε%2
␈β⊃_␈↓ ↓H␈ε"for␈α
an␈α␈y␈αfour␈α
n␈α␈um␈α␈bers␈↓ ∧~␈ε(u␈↓ ∧>␈ε",␈↓ ∧S␈ε(v␈↓ ∧s␈ε",␈↓ ¬λ␈ε(u␈↓ ¬,␈ε",␈↓ ¬A␈ε(v␈↓ ¬l␈ε"with␈α0␈ε6␈α
∀␈↓ ππ␈ε(u␈↓ π5␈ε"<␈↓ πc␈ε(v␈↓ λ
␈ε6∀␈ε"␈α
1,␈α0␈ε6␈α
∀␈↓ 	,␈ε(u␈↓ 	Z␈ε"<␈↓ 
λ␈ε(v␈↓ 
3␈ε6∀␈ε"␈α
1.␈α∂In
␈β⊃$␈↓ ∧/␈ε%1␈↓ ∧d␈ε%1␈↓ ¬≥␈ε%2␈↓ ¬R␈ε%2␈↓ π≤␈ε%1␈↓ πt␈ε%1␈↓ 	A␈ε%2␈↓ 
~␈ε%2
␈β∪(

␈β↓U␈↓ ↓H␈ε"144␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα$␈↓ ↓H␈ε"general,␈α	for␈α
an␈α␈y␈α	positiv␈α␈e␈α	in␈α␈teger␈↓ ¬7␈ε(k␈↓ ¬U␈ε"w␈α␈e␈α	can␈α	require␈α	our␈α
sequence␈α	to␈α	be␈↓ 	`␈ε(k␈↓ 	t␈ε/-distributed
␈βαO␈↓ ↓H␈ε"in␈αthe␈αfollo␈α␈wing␈αsense:
␈ββ≠␈↓ ↓H␈ε2De|nition␈αB.␈ε/␈α→The␈αsequence␈ε"␈α(1)␈ε/␈αis␈αsaid␈αto␈αbe␈↓ π	␈ε(k␈↓ π≥␈ε/-distributed␈αif
␈ββv␈↓ ↓H␈ε"P␈↓ ↓`␈ε"r␈↓ ↓n␈ε"(␈↓ ↓z␈ε(u␈↓ α(␈ε6∀␈↓ αV␈ε(U␈↓ β␈ε"<␈↓ β9␈ε(v␈↓ βZ␈ε",␈↓ βj␈ε".␈αε.␈αε.␈↓ ∧~␈ε",␈↓ ∧*␈ε(u␈↓ ∧Y␈ε6∀␈↓ ¬π␈ε(U␈↓ ε_␈ε"<␈↓ εF␈ε(v␈↓ εh␈ε")␈α
=␈α
(␈↓ π8␈ε(v␈↓ π`␈ε6␈␈↓ λ␈ε(u␈↓ λ0␈ε")␈↓ λB␈ε".␈αε.␈αε.␈↓ λr␈ε"(␈↓ λ}␈ε(v␈↓ 	(␈ε6␈␈↓ 	T␈ε(u␈↓ 	z␈ε")␈↓ α␈ε"(5)
␈β∧α␈↓ α∂␈ε%1␈↓ αo␈ε+n␈↓ βK␈ε%1␈↓ ∧?␈ε+k␈↓ ¬ ␈ε+n␈ε%+␈↓ ¬P␈ε+k␈↓ ¬a␈ε9␈␈ε%␈α␈1␈↓ εW␈ε+k␈↓ πI␈ε%1␈↓ λ!␈ε%1␈↓ 	⊂␈ε+k␈↓ 	i␈ε+k
␈β∧P␈↓ ↓H␈ε/for␈αall␈αch␈α↓oices␈αof␈αreal␈αn␈α␈um␈α␈bers␈↓ ¬-␈ε(u␈↓ ¬Q␈ε/,␈↓ ¬g␈ε(v␈↓ επ␈ε/,␈αwith␈ε"␈α0␈ε6␈α
∀␈↓ π9␈ε(u␈↓ πg␈ε"<␈↓ λ∃␈ε(v␈↓ λ@␈ε6∀␈ε"␈α
1␈ε/,␈αfor␈ε"␈α1␈ε6␈α
∀␈↓ 
_␈ε(j␈↓ 
5␈ε6∀␈↓ 
c␈ε(k␈↓ 
w␈ε/.
␈β∧]␈↓ ¬A␈ε+j␈↓ ¬x␈ε+j␈↓ πN␈ε+j␈↓ λ'␈ε+j
␈β¬≤␈↓ α␈ε"An␈α
equidistributed␈αsequence␈α
is␈α
a␈α1-distributed␈α
sequence.␈α∂Note␈αthat␈α
if␈↓ 
D␈ε(k␈↓ 
b␈ε">␈α
1,
␈β¬G␈↓ ↓H␈ε"a␈↓ ↓f␈ε(k␈↓ ↓z␈ε"-distributed␈αsequence␈αis␈αalw␈α␈a␈α␈ys␈α(␈↓ ¬u␈ε(k␈↓ ε⊃␈ε6␈␈ε"␈αλ1)-distributed,␈α
since␈αw␈α␈e␈αma␈α␈y␈αset␈↓ 
=␈ε(u␈↓ 
l␈ε"=␈α
0
␈β¬S␈↓ 
R␈ε+k
␈β¬r␈↓ ↓H␈ε"and␈↓ α⊂␈ε(v␈↓ α@␈ε"=␈α
1␈α∞in␈α∂Eq.␈α∞(5).␈α⊗Th␈α␈us,␈α∂in␈α∞particular,␈α∂an␈α␈y␈α∞sequence␈α∂that␈α∞is␈α∞kn␈α↓o␈α␈wn␈α∞to␈α∞be
␈β¬␈␈↓ α!␈ε+k
␈βε≥␈↓ ↓H␈ε"4-distributed␈αm␈α␈ust␈αalso␈αbe␈α3-distributed,␈α2-distributed,␈αand␈αequidistributed.␈α∂We
␈βεI␈↓ ↓H␈ε"can␈α
in␈α␈v␈α␈estigate␈α
the␈α
largest␈↓ ∧g␈ε(k␈↓ ¬¬␈ε"for␈α
which␈α
a␈α
giv␈α␈en␈α
sequence␈α
is␈↓ λN␈ε(k␈↓ λb␈ε"-distributed;␈α
and␈α
this
␈βεt␈↓ ↓H␈ε"leads␈αus␈αto␈αform␈α␈ulate
␈βπ?␈↓ ↓H␈ε2De|nition␈αC.␈ε/␈α⊗A␈αsequence␈αis␈αsaid␈αto␈αbe␈ε6␈α1␈ε/-distributed␈↓ λ⊂␈ε/if␈αit␈αis␈↓ λx␈ε(k␈↓ 	␈ε/-distributed␈αfor␈αall
␈βπj␈↓ ↓H␈ε/positiv␈α␈e␈αin␈α␈tegers␈↓ βQ␈ε(k␈↓ βe␈ε/.
␈βλ6␈↓ α␈ε"So␈αfar␈αw␈α␈e␈αha␈α␈v␈α␈e␈αconsidered␈α\␈↓ ¬<␈ε"[␈↓ ¬M␈ε"0,␈αε1)␈αsequences,"␈αi.e.,␈αsequences␈αof␈αreal␈αn␈α␈um␈α␈bers
␈βλa␈↓ ↓H␈ε"lying␈α
bet␈α␈w␈α␈een␈αzero␈α
and␈α
one.␈α∪The␈α
same␈α
ideas␈α
apply␈α
to␈α
in␈α␈teger-v␈α⎇alued␈αsequences;
␈β	␈↓ ↓H␈ε"let␈αus␈αsa␈α␈y␈αa␈αsequence␈ε6␈αh␈↓ ∧"␈ε(X␈↓ ∧S␈ε6i␈ε"␈α
=␈↓ ¬↔␈ε(X␈↓ ¬D␈ε",␈↓ ¬Y␈ε(X␈↓ εε␈ε",␈↓ ε≤␈ε(X␈↓ εI␈ε",␈↓ ε↑␈ε".␈αε.␈αε.␈↓ π∀␈ε"is␈αa␈α\␈↓ πf␈ε(b␈ε"-ary␈αsequence"␈αif␈αeach␈↓ 
Y␈ε(X␈↓ ∀␈ε"is
␈β	→␈↓ ∧@␈ε+n␈↓ ¬5␈ε%0␈↓ ¬w␈ε%1␈↓ ε:␈ε%2␈↓ 
w␈ε+n
␈β	7␈↓ ↓H␈ε"one␈α
of␈αthe␈αin␈α␈tegers␈α
0,␈α1,␈↓ ∧?␈ε".␈αε.␈αε.␈↓ ∧o␈ε",␈ε(␈αb␈ε6␈αε␈␈ε"␈α¬1.␈α⊂Th␈α␈us,␈α
a␈α2-ary␈α(binary)␈α
sequence␈αis␈αa␈α
sequence
␈β	c␈↓ ↓H␈ε"of␈αzeros␈αand␈αones.
␈β
∂␈↓ α␈ε"A␈↓ α4␈ε(k␈↓ αH␈ε"-digit␈α
\␈↓ β;␈ε(b␈ε"␈α↓-ary␈α
n␈α␈um␈α␈ber"␈↓ ¬.␈ε(x␈↓ ¬Q␈ε(x␈↓ ¬z␈ε".␈αε.␈αε.␈↓ ε*␈ε(x␈↓ ε\␈ε"is␈α∞an␈α
ordered␈α∞set␈α
of␈↓ 	≡␈ε(k␈↓ 	@␈ε"in␈α␈tegers,␈α
where
␈β
≤␈↓ ¬B␈ε%1␈↓ ¬e␈ε%2␈↓ ε>␈ε+k
␈β
:␈↓ ↓H␈ε"0␈ε6␈α
∀␈↓ α∩␈ε(x␈↓ α?␈ε"<␈ε(␈α
b␈ε"␈αfor␈α1␈ε6␈α
∀␈↓ ∧␈ε(j␈↓ ∧(␈ε6∀␈↓ ∧V␈ε(k␈↓ ∧j␈ε".
␈β
G␈↓ α&␈ε+j
␈βε␈↓ ↓H␈ε2De|nition␈αD.␈ε/␈α→A␈ε(␈αb␈ε/-ary␈αsequence␈αis␈αsaid␈αto␈αbe␈↓ π∞␈ε(k␈↓ π"␈ε/-distributed␈αif
␈βY␈↓ 	≠␈ε+k
␈βa␈↓ βH␈ε"P␈↓ β`␈ε"r␈↓ βo␈ε"(␈↓ β{␈ε(X␈↓ ∧+␈ε(X␈↓ ¬∂␈ε".␈αε.␈αε.␈↓ ¬?␈ε(X␈↓ εT␈ε"=␈↓ πα␈ε(x␈↓ π%␈ε(x␈↓ πN␈ε".␈αε.␈αε.␈↓ π}␈ε(x␈↓ λ#␈ε")␈α
=␈α
1/␈↓ 	␈ε(b␈↓ α␈ε"(6)
␈βm␈↓ ∧→␈ε+n␈↓ ∧I␈ε+n␈ε%+␈α␈1␈↓ ¬]␈ε+n␈ε%␈α␈+␈↓ ε
␈ε+k␈↓ ε≥␈ε9␈␈ε%1␈↓ π⊗␈ε%1␈↓ π9␈ε%2␈↓ λ∩␈ε+k
␈β<␈↓ ↓H␈ε/for␈αall␈ε(␈αb␈ε/-ary␈αn␈α␈um␈α␈bers␈↓ ∧≥␈ε(x␈↓ ∧@␈ε(x␈↓ ∧i␈ε".␈αε.␈αε.␈↓ ¬→␈ε(x␈↓ ¬>␈ε/.
␈βH␈↓ ∧1␈ε%1␈↓ ∧T␈ε%2␈↓ ¬-␈ε+k
␈β
π␈↓ α␈ε"It␈α⊂is␈α⊂clear␈α⊂from␈α∂this␈α⊂de|nition␈α⊂that␈α⊂if␈↓ εw␈ε(U␈↓ π∨␈ε",␈↓ π:␈ε(U␈↓ πb␈ε",␈↓ π|␈ε".␈αε.␈αε.␈↓ λ6␈ε"is␈α⊂a␈↓ 	␈ε(k␈↓ 	∀␈ε"-distributed␈↓ 
\␈ε"[␈↓ 
l␈ε"0,␈αε1)
␈β
∪␈↓ π⊂␈ε%0␈↓ πR␈ε%1
␈β
2␈↓ ↓H␈ε"sequence,␈α
then␈α∞the␈α
sequence␈ε6␈α∞b␈ε(b␈↓ ¬.␈ε(U␈↓ ¬V␈ε6c␈ε",␈ε6␈α∞b␈ε(b␈↓ ε~␈ε(U␈↓ εB␈ε6c␈ε",␈↓ εh␈ε".␈αε.␈αε.␈↓ π∨␈ε"is␈α∞a␈↓ πd␈ε(k␈↓ πx␈ε"-distributed␈ε(␈α∞b␈ε"-ary␈α
sequence.
␈β
?␈↓ ↓H␈ε↓␈␈↓ ¬G␈ε%0␈↓ ε3␈ε%1␈↓ ≡␈ε↓↓
␈β
]␈↓ ↓V␈ε"If␈α
w␈α␈e␈α∞set␈↓ αk␈ε(u␈↓ β≠␈ε"=␈↓ βL␈ε(x␈↓ βo␈ε"/␈ε(b␈ε",␈↓ ∧)␈ε(v␈↓ ∧V␈ε"=␈α
(␈↓ ¬∪␈ε(x␈↓ ¬?␈ε"+␈α	1)/␈ε(b␈ε",␈↓ εD␈ε(X␈↓ π↓␈ε"=␈ε6␈αb␈ε(b␈↓ πO␈ε(U␈↓ π{␈ε6c␈ε",␈α∞Eq.␈α
(5)␈α∞becomes␈α
Eq.␈α
(6).
␈β
j␈↓ β␈ε+j␈↓ β`␈ε+j␈↓ ∧;␈ε+j␈↓ ¬'␈ε+j␈↓ εb␈ε+n␈↓ πh␈ε+n
␈β∞	␈↓ ↓H␈ε"F␈α⎇urtherm␈α↓ore,␈α⊃ev␈α␈ery␈↓ ∧
␈ε(k␈↓ ∧!␈ε"-distributed␈ε(␈α⊃b␈ε"-ary␈α⊃sequence,␈α⊃for␈↓ λ*␈ε(k␈↓ λP␈ε">␈α⊃1,␈α∩is␈α⊃also␈α⊂(␈↓ 
4␈ε(k␈↓ 
S␈ε6␈␈ε"␈α1)-
␈β∞4␈↓ ↓H␈ε"distributed:␈α
w␈α␈e␈α	add␈α	together␈αλthe␈α	probabilities␈α	for␈α	the␈ε(␈αλb␈ε"␈α↓-ary␈αλn␈α␈um␈α␈bers␈↓ 	a␈ε(x␈↓ 

␈ε".␈αε.␈αε.␈↓ 
:␈ε(x␈↓ ⊂␈ε"0,
␈β∞@␈↓ 	u␈ε%1␈↓ 
N␈ε+k␈↓ 
←␈ε9␈␈ε%␈α␈1
␈β∞←␈↓ ↓H␈ε(x␈↓ ↓q␈ε".␈αε.␈αε.␈↓ α!␈ε(x␈↓ αw␈ε"1,␈↓ β∨␈ε".␈αε.␈αε.␈↓ βO␈ε",␈↓ βe␈ε(x␈↓ ∧∞␈ε".␈αε.␈αε.␈↓ ∧>␈ε(x␈↓ ¬∀␈ε"(␈ε(b␈ε6␈α	␈␈ε"␈αλ1)␈αto␈αobtain
␈β∞k␈↓ ↓\␈ε%1␈↓ α5␈ε+k␈↓ αE␈ε9␈␈ε%1␈↓ βy␈ε%1␈↓ ∧R␈ε+k␈↓ ∧c␈ε9␈␈ε%␈α␈1
␈β∂2␈↓ λV␈ε+k␈↓ λg␈ε9␈␈ε%␈α␈1
␈β∂:␈↓ βW␈ε"P␈↓ βo␈ε"r␈↓ β⎇␈ε"(␈↓ ∧	␈ε(X␈↓ ∧@␈ε".␈αε.␈αε.␈↓ ∧p␈ε(X␈↓ ε¬␈ε"=␈↓ ε3␈ε(x␈↓ ε\␈ε".␈αε.␈αε.␈↓ π␈ε(x␈↓ π↑␈ε")␈α
=␈α
1/␈↓ λF␈ε(b␈↓ 	∪␈ε".
␈β∂F␈↓ ∧'␈ε+n␈↓ ¬∞␈ε+n␈ε%␈α␈+␈↓ ¬>␈ε+k␈↓ ¬N␈ε9␈␈ε%2␈↓ εG␈ε%1␈↓ π ␈ε+k␈↓ π1␈ε9␈␈ε%1
␈β⊂∃␈↓ ↓H␈ε"(Probabilities␈α⊂for␈α⊃disjoin␈α␈t␈α⊃ev␈α␈en␈α␈ts␈α⊂are␈α⊃additiv␈α␈e;␈α∪see␈α⊃exercise␈α⊂5.)␈α"It␈α⊃therefore␈α⊂is
␈β⊂@␈↓ ↓H␈ε"natural␈αto␈αspeak␈αof␈αan␈ε6␈α1␈ε"-distributed␈ε(␈αb␈ε"-ary␈αsequence,␈αas␈αin␈αDe|nition␈αC␈αabo␈α␈v␈α␈e.
␈β⊂m␈↓ α␈ε"The␈α
represen␈α␈tation␈α
of␈αa␈α
positiv␈α␈e␈α
real␈αn␈α␈um␈α␈ber␈α
in␈α
the␈α
radix-␈ε(b␈ε"␈α
n␈α␈um␈α␈ber␈αsystem
␈β⊃_␈↓ ↓H␈ε"ma␈α␈y␈α
be␈αregarded␈α
as␈α
a␈ε(␈α
b␈ε"␈α↓-ary␈αsequence;␈α∞for␈α
example,␈↓ πv␈ε(→␈↓ λ→␈ε"corresponds␈α
to␈α
the␈α10-ary
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"145
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα$␈↓ ↓H␈ε"sequence␈α3,␈α1,␈α4,␈α1,␈α5,␈α9,␈α2,␈α6,␈α5,␈α3,␈α5,␈α8,␈α9,␈↓ ε\␈ε".␈αε.␈αε.␈↓ π∩␈ε".␈α⊂It␈αhas␈αbeen␈αconjectured␈αthat␈αthis
␈βαO␈↓ ↓H␈ε"sequence␈αis␈ε6␈α1␈ε"-distributed,␈αbut␈αn␈α↓obody␈αhas␈αy␈α␈et␈αbeen␈αable␈αto␈αpro␈α␈v␈α␈e␈αthat␈αit␈αis␈αev␈α␈en
␈βαz␈↓ ↓H␈ε"1-distributed.
␈ββ&␈↓ α␈ε"Let␈αus␈α
analyze␈αthese␈αconcepts␈α
a␈αlittle␈αm␈α↓ore␈α
closely␈αin␈αthe␈α
case␈αwhen␈↓ 
.␈ε(k␈↓ 
L␈ε"equals
␈ββQ␈↓ ↓H␈ε"a␈α
million.␈α⊂A␈αbinary␈αsequence␈α
that␈αis␈α1000000-distributed␈αis␈α
going␈αto␈αha␈α␈v␈α␈e␈αruns␈α
of
␈ββ|␈↓ ↓H␈ε"a␈αmillion␈αzeros␈αin␈αa␈αro␈α␈w!␈α⊂Similarly,␈αa␈↓ ε∀␈ε"[␈↓ ε$␈ε"0,␈αε1)␈αsequence␈αthat␈αis␈α1000000-distributed
␈β∧'␈↓ ↓H␈ε"is␈α
going␈α∞to␈α
ha␈α␈v␈α␈e␈α∞runs␈α∞of␈α
a␈α∞million␈α
consecutiv␈α␈e␈α∞v␈α⎇alues␈α∞each␈α
of␈α∞which␈α
is␈α∞less␈α
than
␈β∧L␈↓ εw␈ε%1␈α↓00000␈α↓0
␈β∧N␈↓ ↓L␈ε%1␈↓ εX␈ε%1
␈β∧R␈↓ ↓←␈ε".␈α∃It␈α∞is␈α∞true␈α∞that␈α∞this␈α∞will␈α∞happen␈α
only␈α∞(␈↓ εk␈ε")␈↓ πp␈ε"of␈α∞the␈α
time,␈α∂on␈α∞the␈α
a␈α␈v␈α␈erage,
␈β∧c␈↓ ↓L␈ε%2␈↓ εX␈ε%2
␈β∧g␈↓ ↓L␈∧∧g↓Lα⊂␈↓ εX␈∧∧gεXα⊂
␈β∧}␈↓ ↓H␈ε"but␈αthe␈α
fact␈α
is␈α
that␈α
it␈ε/␈αdoes␈ε"␈α
happen.␈α∪Indeed,␈α
this␈αphen␈α↓omen␈α↓on␈α
will␈α
occur␈α
in␈αan␈α␈y
␈β¬)␈↓ ↓H␈ε"truly␈α
random␈α
sequence,␈α∞using␈α
our␈α
in␈α␈tuitiv␈α␈e␈α∞n␈α↓otion␈α
of␈α
\truly␈α
random."␈α∀One␈α
can
␈β¬T␈↓ ↓H␈ε"easily␈αimagine␈αthat␈αsuch␈αa␈αsituation␈αwill␈αha␈α␈v␈α␈e␈αa␈αdrastic␈αe{ect␈αif␈αthis␈αset␈αof␈αa␈αmil-
␈β¬␈␈↓ ↓H␈ε"lion␈α	\truly␈α
random"␈α	n␈α␈um␈α␈bers␈α
is␈α
being␈α	used␈α
in␈α	a␈α
computer-sim␈α␈ulation␈α	experimen␈α␈t;
␈βε*␈↓ ↓H␈ε"there␈α∞w␈α␈ould␈α∂be␈α∞go␈α↓od␈α∞reason␈α∂to␈α∞complain␈α∂about␈α∞the␈α∂random␈α∞n␈α␈um␈α␈ber␈α∞generator.
␈βεV␈↓ ↓H␈ε"Ho␈α␈w␈α␈ev␈α␈er,␈α∩if␈α⊂w␈α␈e␈α⊃ha␈α␈v␈α␈e␈α⊃a␈α⊂sequence␈α⊃of␈α⊃n␈α␈um␈α␈bers␈α⊂that␈α⊃nev␈α␈er␈α⊃has␈α⊂runs␈α⊃of␈α⊃a␈α⊂million
␈βε|␈↓ ∧l␈ε%1
␈βπ↓␈↓ ↓H␈ε"consecutiv␈α␈e␈↓ β	␈ε(U␈↓ β&␈ε"'s␈α∂less␈α∞than␈↓ ¬␈ε",␈α∂the␈α∂sequence␈α∂is␈α∂n␈α↓ot␈α∂random,␈α∂and␈α∂it␈α∂will␈α∂n␈α↓ot␈α∂be␈α∞a
␈βπ∩␈↓ ∧l␈ε%2
␈βπ∃␈↓ ∧l␈∧π∃∧lα⊂
␈βπ,␈↓ ↓H␈ε"suitable␈αsource␈αof␈αn␈α␈um␈α␈bers␈αfor␈αother␈αconceiv␈α⎇able␈αapplications␈α
that␈αuse␈αextremely
␈βπW␈↓ ↓H␈ε"long␈α
blocks␈α
of␈↓ β1␈ε(U␈↓ βN␈ε"'s␈α
as␈α
input.␈α∪In␈α
summary,␈ε/␈α
a␈α
truly␈α
random␈α
sequence␈α
will␈α
exhibit
␈βλα␈↓ ↓H␈ε/local␈α∞n␈α↓onrandomness␈ε".␈α⊗Local␈α∂n␈α↓onrandomness␈↓ π↔␈ε"is␈α∞necessary␈α∞in␈α∞some␈α∞applications,
␈βλ.␈↓ ↓H␈ε"but␈α∂it␈α∂is␈α∂disastrous␈α⊂in␈α∂others.␈α~We␈α∂are␈α∂forced␈α∂to␈α∂conclude␈α⊂that␈α∂n␈α↓o␈α∂sequence␈α∂of
␈βλY␈↓ ↓H␈ε"\random"␈αn␈α␈um␈α␈bers␈αcan␈αbe␈αadequate␈αfor␈αev␈α␈ery␈αapplication.
␈β	∧␈↓ α␈ε"In␈α∞a␈α∂similar␈α∞v␈α␈ein,␈α∂one␈α∂ma␈α␈y␈α∞argue␈α∞that␈α∂there␈α∞is␈α∂n␈α↓o␈α∞w␈α␈a␈α␈y␈α∞to␈α∂judge␈α∞whether␈α∞a
␈β	/␈↓ ↓H␈ε/|nite␈ε"␈αsequence␈αis␈αrandom␈αor␈αn␈α↓ot;␈αan␈α␈y␈αparticular␈↓ π>␈ε"sequence␈αis␈αjust␈αas␈αlik␈α␈ely␈αas␈αan␈α␈y
␈β	Z␈↓ ↓H␈ε"other␈α
one.␈α∃These␈α∞facts␈α∞are␈α
de|nitely␈α∞stum␈α␈bling␈α∞blocks␈α
if␈α∞w␈α␈e␈α∞are␈α
ev␈α␈er␈α∞to␈α∞ha␈α␈v␈α␈e␈α
a
␈β
ε␈↓ ↓H␈ε"useful␈α∞de|nition␈α∂of␈α∞randomness,␈α⊂but␈α∞they␈α∂are␈α∂n␈α↓ot␈α∞really␈α∂cause␈α∞for␈α∂alarm.␈α_It␈α∞is
␈β
1␈↓ ↓H␈ε"still␈αpossible␈αto␈αgiv␈α␈e␈αa␈αde|nition␈αfor␈αthe␈αrandomness␈αof␈αin|nite␈αsequences␈αof␈αreal
␈β
\␈↓ ↓H␈ε"n␈α␈um␈α␈bers␈α
in␈αsuch␈αa␈α
w␈α␈a␈α␈y␈αthat␈αthe␈α
corresponding␈αtheory␈α
(view␈α␈ed␈αproperly)␈αwill␈α
giv␈α␈e
␈βπ␈↓ ↓H␈ε"us␈α∂a␈α⊂great␈α⊂deal␈α⊂of␈α∂insigh␈α␈t␈α⊂concerning␈α⊂the␈α⊂ordinary␈α∂|nite␈α⊂sequences␈α⊂of␈α∂rational
␈β2␈↓ ↓H␈ε"n␈α␈um␈α␈bers␈α
that␈α
are␈α
actually␈α
generated␈αon␈α
a␈α
computer.␈α∪F␈α⎇urtherm␈α↓ore,␈α∞w␈α␈e␈αshall␈α
see
␈β↑␈↓ ↓H␈ε"later␈α
in␈αthis␈α
section␈αthat␈α
there␈αare␈αsev␈α␈eral␈α
plausible␈αde|nitions␈α
of␈αrandomness␈α
for
␈β	␈↓ ↓H␈ε"|nite␈αsequences.
␈βI␈↓ ↓H␈ε2B.␈ε6␈α
1␈ε2-distributed␈α
sequences.␈ε"␈α≠Let␈α
us␈α
n␈α↓o␈α␈w␈α∞undertak␈α␈e␈α
a␈α
brief␈α
study␈α
of␈α
the␈α
theory
␈βt␈↓ ↓H␈ε"of␈αsequences␈αthat␈αare␈ε6␈α1␈ε"-distributed.␈α⊂T␈α⎇o␈αdescribe␈αthe␈αtheory␈αadequately,␈αw␈α␈e␈αwill
␈β
∨␈↓ ↓H␈ε"need␈αto␈α
use␈αa␈α
bit␈αof␈α
higher␈αmathematics,␈α
so␈αw␈α␈e␈α
assume␈αin␈α
the␈αremainder␈α
of␈αthis
␈β
J␈↓ ↓H␈ε"subsection␈α
that␈α
the␈α
reader␈αkn␈α↓o␈α␈ws␈α
the␈α
material␈α
ordinarily␈αtaugh␈α␈t␈α
in␈α
an␈α
\adv␈α⎇anced
␈β
v␈↓ ↓H␈ε"calculus"␈αcourse.
␈β∞!␈↓ α␈ε"First␈α∂it␈α∞is␈α∂con␈α␈v␈α␈enien␈α␈t␈α∂to␈α∂generalize␈α∞De|nition␈α∂A␈↓ λ¬␈ε",␈α∂since␈α∂the␈α∂limit␈α∞appearing
␈β∞L␈↓ ↓H␈ε"there␈αdoes␈αn␈α↓ot␈αexist␈αfor␈αall␈αsequences.␈α⊂Let␈αus␈αde|ne
␈β∂α␈↓ αx␈ε↓␈␈↓ βM␈ε↓↓␈↓ ¬π␈ε↓␈␈↓ ε␈ε↓↓␈↓ π␈ε↓␈␈↓ πa␈ε↓↓␈↓ 	⊃␈ε↓␈␈↓ 

␈ε↓↓
␈β∂!␈↓ αR␈∧∂!αRα'␈↓ αR␈ε"P␈↓ αj␈ε"r␈↓ βε␈ε(S␈↓ β ␈ε"(␈ε(n␈ε"␈α␈)␈↓ βe␈ε"=␈↓ ∧∪␈ε"lim␈αεsup␈↓ ¬∃␈ε(↔␈↓ ¬*␈ε"(␈ε(n␈ε")/␈ε(n␈↓ ε∞␈ε",␈↓ εf␈ε"P␈↓ ε}␈ε"r␈↓ π~␈ε(S␈↓ π3␈ε"(␈ε(n␈ε")␈↓ πy␈ε"=␈↓ λ'␈ε"lim␈αεin␈↓ λ⎇␈ε"f␈↓ 	∨␈ε(↔␈↓ 	4␈ε"(␈ε(n␈ε")/␈ε(n␈↓ 
_␈ε".␈↓ α␈ε"(7)
␈β∂A␈↓ λ2␈ε+n␈ε9!␈α␈1
␈β∂B␈↓ εf␈∧∂Bεfα'
␈β∂H␈↓ ∧#␈ε+n␈ε9!1
␈β∂t␈↓ αL␈ε↓␈␈↓ β!␈ε↓↓␈↓ λ_␈ε↓␈␈↓ λm␈ε↓↓␈↓ 	s␈ε↓␈␈↓ 
H␈ε↓↓
␈β⊂∪␈↓ ↓H␈ε"Then␈↓ α&␈ε"P␈↓ α>␈ε"r␈↓ αZ␈ε(S␈↓ αs␈ε"(␈ε(n␈ε")␈↓ β/␈ε",␈αif␈αit␈αexists,␈αis␈αthe␈αcomm␈α↓on␈αv␈α⎇alue␈αof␈↓ πr␈ε"P␈↓ λ
␈ε"r␈↓ λ&␈ε(S␈↓ λ?␈ε"(␈ε(n␈ε")␈↓ 	π␈ε"and␈↓ 	M␈∧⊂∪	Mα'␈↓ 	M␈ε"P␈↓ 	e␈ε"r␈↓ 
↓␈ε(S␈↓ 
~␈ε"(␈ε(n␈ε")␈↓ 
V␈ε".
␈β⊂4␈↓ πr␈∧⊂4πrα'
␈β⊂B␈↓ α␈ε"We␈α∞ha␈α␈v␈α␈e␈α∂seen␈α∞that␈α∂a␈↓ ∧e␈ε(k␈↓ ∧y␈ε"-distributed␈↓ ε@␈ε"[␈↓ εP␈ε"0,␈αε1)␈α∂sequence␈α∞leads␈α∞to␈α∂a␈↓ 	`␈ε(k␈↓ 	t␈ε"-distributed
␈β⊂m␈↓ ↓H␈ε(b␈ε"-ary␈↓ α!␈ε"sequence,␈α
if␈↓ β\␈ε(U␈↓ ∧β␈ε"is␈α
replaced␈α	by␈ε6␈α
b␈ε(b␈↓ ¬␈␈ε(U␈↓ ε≤␈ε6c␈ε".␈α⊂Our␈α	|rst␈α
theorem␈α
sh␈α↓o␈α␈ws␈α	that␈α
a␈α	con␈α␈v␈α␈erse
␈β⊃_␈↓ ↓H␈ε"result␈↓ α,␈ε"is␈αalso␈αtrue.
␈β∪(

␈β↓U␈↓ ↓H␈ε"146␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα"␈↓ ↓H␈ε2Theorem␈αA.␈ε/␈α→Let␈ε6␈αh␈↓ βk␈ε(U␈↓ ∧⊗␈ε6i␈ε"␈α
=␈↓ ∧Z␈ε(U␈↓ ¬α␈ε/,␈↓ ¬_␈ε(U␈↓ ¬@␈ε/,␈↓ ¬V␈ε(U␈↓ ¬}␈ε/,␈↓ ε∀␈ε".␈αε.␈αε.␈↓ εJ␈ε/be␈αa␈↓ π_␈ε"[␈↓ π(␈ε"0,␈αε1)␈ε/␈αsequence.␈α⊂If␈αthe␈αsequence
␈βα/␈↓ ∧∧␈ε+n␈↓ ∧s␈ε%0␈↓ ¬1␈ε%1␈↓ ¬o␈ε%2
␈βαr␈↓ ∧⊃␈ε6hb␈↓ ∧+␈ε(b␈↓ ∧J␈ε(U␈↓ ∧u␈ε6ci␈ε"␈α
=␈ε6␈α
b␈↓ ¬U␈ε(b␈↓ ¬u␈ε(U␈↓ ε≥␈ε6c␈ε",␈ε6␈α∞b␈↓ εQ␈ε(b␈↓ εp␈ε(U␈↓ π_␈ε6c␈ε",␈ε6␈α∞b␈↓ πL␈ε(b␈↓ πl␈ε(U␈↓ λ∀␈ε6c␈ε",␈↓ λ:␈ε".␈αε.␈αε.
␈βα}␈↓ ∧;␈ε+j␈↓ ∧c␈ε+n␈↓ ¬f␈ε+j␈↓ ε∞␈ε%0␈↓ εa␈ε+j␈↓ π	␈ε%1␈↓ π\␈ε+j␈↓ λ∧␈ε%2
␈ββA␈↓ ↓H␈ε/is␈α∂a␈↓ α⊂␈ε(k␈↓ α$␈ε/-distributed␈↓ βk␈ε(b␈↓ ∧␈ε/-ary␈α∂sequence␈α∂for␈α⊂all␈↓ εa␈ε(b␈↓ π∂␈ε/in␈α⊂an␈α∂in|nite␈α∂sequence␈α∂of␈α∂in␈α␈tegers
␈ββM␈↓ β|␈ε+j␈↓ εq␈ε+j
␈ββl␈↓ ↓H␈ε"1␈α
<␈↓ α∩␈ε(b␈↓ α;␈ε"<␈↓ αi␈ε(b␈↓ β∪␈ε"<␈↓ βA␈ε(b␈↓ βj␈ε"<␈↓ ∧_␈ε6↓␈αε↓␈αε↓␈↓ ∧H␈ε/,␈αthen␈αthe␈αoriginal␈αsequence␈ε6␈αh␈↓ λ⊂␈ε(U␈↓ λ;␈ε6i␈ε/␈αis␈↓ λw␈ε(k␈↓ 	␈ε/-distributed.
␈ββx␈↓ α"␈ε%1␈↓ αy␈ε%2␈↓ βQ␈ε%3␈↓ λ)␈ε+n
␈β∧$␈↓ λ9␈ε+j␈↓ 
]␈ε+j
␈β∧*␈↓ α␈ε"As␈αan␈α
example␈αof␈αthis␈α
theorem,␈αsuppose␈αthat␈↓ πP␈ε(b␈↓ πy␈ε"=␈↓ λ'␈ε"2␈↓ λH␈ε".␈α⊂The␈αsequence␈ε6␈α
b␈↓ 
K␈ε"2␈↓ 
l␈ε(U␈↓ ∀␈ε6c␈ε",
␈β∧6␈↓ π`␈ε+j␈↓ ¬␈ε%0
␈β∧O␈↓ ↓h␈ε+j
␈β∧U␈↓ ↓H␈ε6b␈↓ ↓V␈ε"2␈↓ ↓w␈ε(U␈↓ α∨␈ε6c␈ε",␈↓ αI␈ε".␈αε.␈αε.␈↓ β∧␈ε"is␈α⊃essen␈α␈tially␈α⊃the␈α⊂sequence␈α⊃of␈α⊃the␈α⊃|rst␈↓ πz␈ε(j␈↓ λ≡␈ε"bits␈α⊃of␈α⊃the␈α⊃binary␈α⊂repre-
␈β∧a␈↓ α⊂␈ε%1
␈β¬␈↓ ↓H␈ε"sen␈α␈tations␈αof␈↓ β≠␈ε(U␈↓ βC␈ε",␈↓ βY␈ε(U␈↓ ∧↓␈ε",␈↓ ∧↔␈ε".␈αε.␈αε.␈↓ ∧M␈ε".␈α⊂If␈αall␈αthese␈αin␈α␈teger␈αsequences␈αare␈↓ λn␈ε(k␈↓ 	α␈ε"-distributed,␈αin␈αthe
␈β¬␈↓ β3␈ε%0␈↓ βq␈ε%1
␈β¬+␈↓ ↓H␈ε"sense␈αof␈α
De|nition␈αD␈↓ ∧∪␈ε",␈α
then␈αthe␈α
real-v␈α⎇alued␈αsequence␈↓ λε␈ε(U␈↓ λ.␈ε",␈↓ λE␈ε(U␈↓ λm␈ε",␈↓ 	β␈ε".␈αε.␈αε.␈↓ 	:␈ε"m␈α␈ust␈αalso␈α
be␈↓ ␈ε(k␈↓  ␈ε"-
␈β¬8␈↓ λ∨␈ε%0␈↓ λ]␈ε%1
␈β¬V␈↓ ↓H␈ε"distributed␈αin␈αthe␈αsense␈αof␈αDe|nition␈αB.
␈βε∀␈↓ ↓H␈ε/Pro␈α↓of␈α
of␈αTheorem␈αA.␈ε"␈α∃If␈α
the␈αsequence␈ε6␈αb␈ε(b␈↓ ε2␈ε(U␈↓ εZ␈ε6c␈ε",␈ε6␈αb␈ε(b␈↓ π≠␈ε(U␈↓ πC␈ε6c␈ε",␈↓ πf␈ε".␈αε.␈αε.␈↓ λ≠␈ε"is␈↓ λ>␈ε(k␈↓ λR␈ε"-distributed,␈αit␈α
follo␈α␈ws
␈βε!␈↓ εK␈ε%0␈↓ π4␈ε%1
␈βε?␈↓ ↓H␈ε"by␈αthe␈αaddition␈αof␈αprobabilities␈αthat␈α
Eq.␈α(5)␈αh␈α↓olds␈αwhenev␈α␈er␈αeach␈↓ 	S␈ε(u␈↓ 
β␈ε"and␈↓ 
I␈ε(v␈↓ 
v␈ε"is␈αa
␈βεL␈↓ 	h␈ε+j␈↓ 
[␈ε+j
␈βεk␈↓ ↓H␈ε"rational␈α
n␈α␈um␈α␈ber␈α
with␈α
den␈α↓ominator␈ε(␈α
b␈ε"␈α↓.␈α∪No␈α␈w␈α
let␈↓ π0␈ε(u␈↓ πT␈ε",␈↓ πl␈ε(v␈↓ λ~␈ε"be␈α
an␈α␈y␈α
real␈α
n␈α␈um␈α␈bers,␈α
and
␈βεw␈↓ πE␈ε+j␈↓ π⎇␈ε+j
␈βπ⊂␈↓ α⊂␈ε90␈↓ αH␈ε90
␈βπ⊗␈↓ ↓H␈ε"let␈↓ ↓|␈ε(u␈↓ α ␈ε",␈↓ α6␈ε(v␈↓ αd␈ε"be␈αrational␈αn␈α␈um␈α␈bers␈αwith␈αden␈α↓ominator␈ε(␈αb␈ε"␈αsuch␈αthat
␈βπ&␈↓ α⊂␈ε+j␈↓ αH␈ε+j
␈βπa␈↓ β]␈ε90␈↓ ¬∃␈ε90␈↓ εw␈ε90␈↓ λ*␈ε90
␈βπi␈↓ βH␈ε(u␈↓ βv␈ε6∀␈↓ ∧$␈ε(u␈↓ ∧R␈ε"<␈↓ ¬␈ε(u␈↓ ¬,␈ε"+␈αλ1/␈ε(b␈ε"␈α↓,␈↓ εe␈ε(v␈↓ π⊃␈ε6∀␈↓ π?␈ε(v␈↓ πi␈ε"<␈↓ λ↔␈ε(v␈↓ λA␈ε"+␈αλ1/␈ε(b␈ε"␈α↓.
␈βπu␈↓ ∧9␈ε+j␈↓ πP␈ε+j
␈βπz␈↓ β]␈ε+j␈↓ ¬∃␈ε+j␈↓ εw␈ε+j␈↓ λ*␈ε+j
␈βλ8␈↓ ↓H␈ε"Let␈↓ α␈ε(S␈↓ α$␈ε"(␈ε(n␈ε")␈α
be␈α∞the␈α
statemen␈α␈t␈α∞that␈↓ ¬H␈ε(u␈↓ ¬y␈ε6∀␈↓ ε)␈ε(U␈↓ εa␈ε"<␈↓ π∩␈ε(v␈↓ π2␈ε",␈↓ πJ␈ε".␈αε.␈αε.␈↓ πz␈ε",␈↓ λ∩␈ε(u␈↓ λD␈ε6∀␈↓ λu␈ε(U␈↓ 
π␈ε"<␈↓ 
8␈ε(v␈↓ 
Z␈ε".␈α∀We
␈βλE␈↓ ¬]␈ε%1␈↓ εB␈ε+n␈↓ π#␈ε%1␈↓ λ'␈ε+k␈↓ 	
␈ε+n␈ε%+␈↓ 	=␈ε+k␈↓ 	N␈ε9␈␈ε%1␈↓ 
I␈ε+k
␈βλc␈↓ ↓H␈ε"ha␈α␈v␈α␈e
␈β	!␈↓ α∩␈ε↓␈␈↓ αg␈ε↓↓
␈β	&␈↓ βY␈ε↓∩␈↓ 
⊂␈ε↓∪
␈β	'␈↓ ελ␈ε"1␈↓ 	z␈ε"1
␈β	7␈↓ ∧∧␈ε90␈↓ ¬A␈ε90␈↓ π↔␈ε90␈↓ 	1␈ε90
␈β	?␈↓ ↓l␈ε"P␈↓ α∧␈ε"r␈↓ α ␈ε(S␈↓ α9␈ε"(␈ε(n␈ε")␈↓ α␈␈ε6∀␈↓ β-␈ε"P␈↓ βE␈ε"r␈↓ βo␈ε(u␈↓ ∧≥␈ε6∀␈↓ ∧K␈ε(U␈↓ ¬␈ε"<␈↓ ¬.␈ε(v␈↓ ¬X␈ε"+␈↓ ε≡␈ε",␈↓ ε8␈ε".␈αε.␈αε.␈↓ εh␈ε",␈↓ πα␈ε(u␈↓ π2␈ε6∀␈↓ π`␈ε(U␈↓ λp␈ε"<␈↓ 	≡␈ε(v␈↓ 	J␈ε"+
␈β	@␈↓ ↓l␈∧	@↓lα'
␈β	L␈↓ ∧d␈ε+n␈↓ πy␈ε+n␈ε%␈α␈+␈↓ λ)␈ε+k␈↓ λ9␈ε9␈␈ε%1
␈β	P␈↓ ∧∧␈ε%1␈↓ ¬A␈ε%1␈↓ π↔␈ε+k␈↓ 	1␈ε+k
␈β	S␈↓ ελ␈∧	Sελα∩␈↓ 	z␈∧	S	zα∩
␈β	Y␈↓ ε	␈ε(b␈↓ 	z␈ε(b
␈β
∧␈↓ ε↑␈ε↓∩␈↓ λ<␈ε↓∪␈↓ 	λ␈ε↓∩␈↓ 
h␈ε↓∪
␈β
¬␈↓ λ&␈ε"1␈↓ 
R␈ε"1
␈β
∃␈↓ ππ␈ε90␈↓ π↑␈ε90␈↓ 	1␈ε90␈↓ 

␈ε90
␈β
≥␈↓ ε0␈ε"=␈↓ εt␈ε(v␈↓ π≡␈ε6␈␈↓ πJ␈ε(u␈↓ πv␈ε"+␈↓ λX␈ε".␈αε.␈αε.␈↓ 	≡␈ε(v␈↓ 	I␈ε6␈␈↓ 	u␈ε(u␈↓ 
"␈ε"+␈↓ 
}␈ε";
␈β
.␈↓ ππ␈ε%1␈↓ π↑␈ε%1␈↓ 	1␈ε+k␈↓ 

␈ε+k
␈β
1␈↓ λ&␈∧
1λ&α∩␈↓ 
R␈∧
1
Rα∩
␈β
7␈↓ λ'␈ε(b␈↓ 
S␈ε(b
␈β
i␈↓ α∩␈ε↓␈␈↓ αg␈ε↓↓
␈β
n␈↓ βY␈ε↓∩␈↓ 
⊂␈ε↓∪
␈β
o␈↓ ∧K␈ε"1␈↓ π`␈ε"1
␈β
␈␈↓ ∧∧␈ε90␈↓ ε∂␈ε90␈↓ π↔␈ε90␈↓ 	␈␈ε90
␈βπ␈↓ ↓l␈ε"P␈↓ α∧␈ε"r␈↓ α ␈ε(S␈↓ α9␈ε"(␈ε(n␈ε")␈↓ α␈␈ε6∃␈↓ β-␈ε"P␈↓ βE␈ε"r␈↓ βo␈ε(u␈↓ ∧≠␈ε"+␈↓ ∧k␈ε6∀␈↓ ¬→␈ε(U␈↓ ¬N␈ε"<␈↓ ¬|␈ε(v␈↓ ε≡␈ε",␈↓ ε8␈ε".␈αε.␈αε.␈↓ εh␈ε",␈↓ πα␈ε(u␈↓ π0␈ε"+␈↓ λ␈ε6∀␈↓ λ.␈ε(U␈↓ 	>␈ε"<␈↓ 	l␈ε(v
␈β∀␈↓ ¬2␈ε+n␈↓ λG␈ε+n␈ε%␈α␈+␈↓ λw␈ε+k␈↓ 	π␈ε9␈␈ε%1
␈β_␈↓ ∧∧␈ε%1␈↓ ε∂␈ε%1␈↓ π↔␈ε+k␈↓ 	␈␈ε+k
␈β≠␈↓ ∧K␈∧≠∧Kα∩␈↓ π`␈∧≠π`α∩
␈β!␈↓ ∧L␈ε(b␈↓ πa␈ε(b
␈β)␈↓ ↓l␈∧)↓lα'
␈βL␈↓ ε↑␈ε↓∩␈↓ λ<␈ε↓∪␈↓ 	λ␈ε↓∩␈↓ 
h␈ε↓∪
␈βM␈↓ λ&␈ε"1␈↓ 
R␈ε"1
␈β]␈↓ ππ␈ε90␈↓ π↑␈ε90␈↓ 	1␈ε90␈↓ 

␈ε90
␈βe␈↓ ε0␈ε"=␈↓ εt␈ε(v␈↓ π≡␈ε6␈␈↓ πJ␈ε(u␈↓ πv␈ε6␈␈↓ λX␈ε".␈αε.␈αε.␈↓ 	≡␈ε(v␈↓ 	I␈ε6␈␈↓ 	u␈ε(u␈↓ 
"␈ε6␈␈↓ 
}␈ε".
␈βv␈↓ ππ␈ε%1␈↓ π↑␈ε%1␈↓ 	1␈ε+k␈↓ 

␈ε+k
␈βy␈↓ λ&␈∧yλ&α∩␈↓ 
R␈∧y
Rα∩
␈β␈␈↓ λ'␈ε(b␈↓ 
S␈ε(b
␈βD␈↓ αG␈ε90␈↓ β$␈ε90
␈βJ␈↓ ↓H␈ε"No␈α␈w␈ε6␈α∂j␈ε"(␈↓ α4␈ε(v␈↓ αa␈ε6␈␈↓ β∂␈ε(u␈↓ β=␈ε6ε␈ε"␈α1/␈ε(b␈ε")␈ε6␈α
␈␈ε"␈α(␈↓ ∧q␈ε(v␈↓ ¬≤␈ε6␈␈↓ ¬J␈ε(u␈↓ ¬n␈ε")␈ε6j␈α⊂∀␈ε"␈α⊂2/␈ε(b␈ε";␈α∩since␈α∂our␈α⊂inequalities␈α∂h␈α↓old␈α⊂for␈α∂all
␈βV␈↓ ¬α␈ε+j␈↓ ¬←␈ε+j
␈β[␈↓ αG␈ε+j␈↓ β$␈ε+j
␈βu␈↓ ↓H␈ε(b␈ε"␈α
=␈↓ α⊂␈ε(b␈↓ α/␈ε",␈αand␈αsince␈↓ βc␈ε(b␈↓ ∧
␈ε6!␈α
1␈ε"␈αas␈↓ ¬↔␈ε(j␈↓ ¬4␈ε6!␈α
1␈ε",␈αw␈α␈e␈αha␈α␈v␈α␈e
␈β
↓␈↓ α ␈ε+j␈↓ βt␈ε+j
␈β
*␈↓ ¬→␈ε↓␈␈↓ ¬n␈ε↓↓␈↓ ε[␈ε↓␈␈↓ π/␈ε↓↓
␈β
H␈↓ ↓a␈ε"(␈↓ ↓m␈ε(v␈↓ α⊗␈ε6␈␈↓ αB␈ε(u␈↓ αf␈ε")␈↓ αx␈ε".␈αε.␈αε.␈↓ β(␈ε"(␈↓ β4␈ε(v␈↓ β↑␈ε6␈␈↓ ∧
␈ε(u␈↓ ∧/␈ε")␈ε6␈α
∀␈↓ ∧s␈ε"P␈↓ ¬␈ε"r␈↓ ¬'␈ε(S␈↓ ¬@␈ε"(␈ε(n␈ε")␈↓ εε␈ε6∀␈↓ ε4␈∧
Hε4α'␈↓ ε4␈ε"P␈↓ εL␈ε"r␈↓ εi␈ε(S␈↓ πα␈ε"(␈ε(n␈ε"␈α␈)␈↓ πG␈ε6∀␈ε"␈α
(␈↓ λ↓␈ε(v␈↓ λ*␈ε6␈␈↓ λV␈ε(u␈↓ λz␈ε")␈↓ 	␈ε".␈αε.␈αε.␈↓ 	<␈ε"(␈↓ 	H␈ε(v␈↓ 	r␈ε6␈␈↓ 
≡␈ε(u␈↓ 
C␈ε").
␈β
Q␈↓ ↓␈∧
Q↓≠∂
␈β
U␈↓ ↓␈␈ε%1␈↓ αW␈ε%1␈↓ βE␈ε+k␈↓ ∧∨␈ε+k␈↓ λ∪␈ε%1␈↓ λk␈ε%1␈↓ 	Z␈ε+k␈↓ 
3␈ε+k
␈β
j␈↓ ∧s␈∧
j∧sα'
␈β∞↔␈↓ α␈ε"The␈α∞next␈α∂theorem␈α∞is␈α∞our␈α∂main␈α∞to␈α↓ol␈α∞for␈α∞pro␈α␈ving␈α∂things␈α∞about␈↓ 	`␈ε(k␈↓ 	t␈ε"-distributed
␈β∞C␈↓ ↓H␈ε"sequences.
␈β∂␈↓ ↓H␈ε2Theorem␈α
B.␈ε/␈α∀Let␈ε6␈αh␈↓ βa␈ε(U␈↓ ∧␈ε6i␈ε/␈αbe␈αa␈↓ ∧n␈ε(k␈↓ ¬α␈ε/-distributed␈↓ εE␈ε"[␈↓ εU␈ε"0,␈αε1)␈ε/␈αsequence,␈αand␈α
let␈↓ 	4␈ε(f␈↓ 	I␈ε"(␈↓ 	U␈ε(x␈↓ 	y␈ε",␈↓ 
	␈ε(x␈↓ 
,␈ε",␈↓ 
<␈ε".␈αε.␈αε.␈↓ 
l␈ε",␈↓ 
|␈ε(x␈↓  ␈ε")
␈β∂
␈↓ βz␈ε+n␈↓ 	i␈ε%1␈↓ 
≥␈ε%2␈↓ ⊂␈ε+k
␈β∂,␈↓ ↓H␈ε/be␈αa␈αRiemann-in␈α␈tegrable␈αfunction␈αof␈↓ ε∂␈ε(k␈↓ ε/␈ε/v␈α⎇ariables;␈↓ πM␈ε/then
␈β∂g␈↓ ∧i␈ε↓X
␈β∂q␈↓ ∧/␈ε"1
␈β⊂	␈↓ βc␈ε"lim␈↓ ¬?␈ε(f␈↓ ¬U␈ε"(␈↓ ¬a␈ε(U␈↓ ε	␈ε",␈↓ ε→␈ε(U␈↓ εm␈ε",␈↓ ε⎇␈ε".␈αε.␈αε.␈↓ π-␈ε",␈↓ π=␈ε(U␈↓ λ@␈ε")
␈β⊂⊗␈↓ ¬y␈ε+j␈↓ ε1␈ε+j␈↓ εA␈ε%+␈α␈1␈↓ πV␈ε+j␈↓ πe␈ε%+␈↓ λβ␈ε+k␈↓ λ∪␈ε9␈␈ε%1
␈β⊂≡␈↓ ∧-␈∧⊂≡∧-α⊗
␈β⊂#␈↓ ∧-␈ε(n
␈β⊂*␈↓ βV␈ε+n␈ε9␈α␈!1
␈β⊂:␈↓ ∧M␈ε%0␈ε9∀␈↓ ∧z␈ε+j␈↓ ¬	␈ε%<␈ε+n
␈β⊂F␈↓ α␈ε"(8)
␈β⊂W␈↓ ∧P␈ε↓Z␈↓ ¬?␈ε↓Z
␈β⊂]␈↓ ∧t␈ε%1␈↓ ¬c␈ε%1
␈β⊃β␈↓ ∧"␈ε"=␈↓ ¬	␈ε6↓␈αε↓␈αε↓␈↓ ¬x␈ε(f␈↓ ε
␈ε"(␈↓ ε→␈ε(x␈↓ ε=␈ε",␈↓ εM␈ε(x␈↓ εp␈ε",␈↓ π␈ε".␈αε.␈αε.␈↓ π0␈ε",␈↓ π@␈ε(x␈↓ πd␈ε")␈ε(␈αεd␈↓ λ
␈ε(x␈↓ λ3␈ε".␈αε.␈αε.␈↓ λc␈ε(d␈↓ λv␈ε(x␈↓ 	≠␈ε".
␈β⊃⊂␈↓ ε-␈ε%1␈↓ εa␈ε%2␈↓ πT␈ε+k␈↓ λ≡␈ε%1␈↓ 	
␈ε+k
␈β⊃&␈↓ ∧d␈ε%0␈↓ ¬S␈ε%0
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"147
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα$␈↓ ↓H␈ε/Pro␈α↓of.␈ε"␈α~The␈α
de|nition␈α
of␈α
a␈↓ ∧y␈ε(k␈↓ ¬
␈ε"-distributed␈α
sequence␈αstates␈α
that␈α
this␈α
result␈α
is␈αtrue
␈βαO␈↓ ↓H␈ε"in␈αthe␈αspecial␈αcase␈αthat
␈ββ ␈↓ ∧x␈ε"1,␈↓ ¬\␈ε"if␈↓ ¬}␈ε(u␈↓ ε,␈ε6∀␈↓ εZ␈ε(x␈↓ ππ␈ε"<␈↓ π5␈ε(v␈↓ πV␈ε",␈↓ πf␈ε".␈αε.␈αε.␈↓ λ⊗␈ε",␈↓ λ&␈ε(u␈↓ λU␈ε6∀␈↓ 	β␈ε(x␈↓ 	1␈ε"<␈↓ 	←␈ε(v␈↓ 
α␈ε";
␈ββ!␈↓ ∧b␈ε↓~
␈ββ,␈↓ ε∩␈ε%1␈↓ εn␈ε%1␈↓ πF␈ε%1␈↓ λ:␈ε+k␈↓ 	↔␈ε+k␈↓ 	q␈ε+k
␈ββ;␈↓ αe␈ε(f␈↓ αz␈ε"(␈↓ βε␈ε(x␈↓ β)␈ε",␈↓ β9␈ε".␈αε.␈αε.␈↓ βi␈ε",␈↓ βy␈ε(x␈↓ ∧≡␈ε")␈α
=␈↓ α␈ε"(9)
␈ββG␈↓ β~␈ε%1␈↓ ∧
␈ε+k
␈ββV␈↓ ∧x␈ε"0,␈↓ ¬\␈ε"otherwise.
␈β∧(␈↓ ↓H␈ε"Therefore␈α
Eq.␈α	(8)␈α
is␈α
true␈α
whenev␈α␈er␈↓ ¬c␈ε(f␈↓ εα␈ε"=␈↓ ε0␈ε(a␈↓ εR␈ε(f␈↓ εw␈ε"+␈↓ π∨␈ε(a␈↓ πA␈ε(f␈↓ πf␈ε"+␈↓ λ
␈ε6↓␈αε↓␈αε↓␈↓ λ;␈ε"+␈↓ λc␈ε(a␈↓ 	⊂␈ε(f␈↓ 	G␈ε"and␈α
when␈α	each
␈β∧4␈↓ εC␈ε%1␈↓ εd␈ε%1␈↓ π2␈ε%2␈↓ πS␈ε%2␈↓ λv␈ε+m␈↓ 	"␈ε+m
␈β∧S␈↓ ↓H␈ε(f␈↓ ↓u␈ε"is␈αa␈αfunction␈αof␈αt␈α␈ype␈α(9);␈αin␈αother␈αw␈α␈ords,␈αEq.␈α(8)␈αh␈α↓olds␈αwhenev␈α␈er␈↓ 	k␈ε(f␈↓ 
␈ε"is␈αa␈α\step-
␈β∧`␈↓ ↓Y␈ε+j
␈β∧}␈↓ ↓H␈ε"function"␈αobtained␈α
by␈α(i)␈α
partitioning␈αthe␈αunit␈↓ π0␈ε(k␈↓ πD␈ε"-dimensional␈α
cube␈αin␈α␈to␈αsubcells
␈β¬*␈↓ ↓H␈ε"wh␈α↓ose␈α⊂faces␈α⊂are␈α⊃parallel␈α⊂to␈α⊃the␈α⊂co␈α↓ordinate␈α⊂axes,␈α∩and␈α⊂(ii)␈α⊃assigning␈α⊂a␈α⊂constan␈α␈t
␈β¬U␈↓ ↓H␈ε"v␈α⎇alue␈αto␈↓ αP␈ε(f␈↓ αq␈ε"on␈αeach␈αsubcell.
␈βε␈↓ α␈ε"No␈α␈w␈αlet␈↓ β∩␈ε(f␈↓ β3␈ε"be␈αan␈α␈y␈αRiemann-in␈α␈tegrable␈αfunction.␈α⊂If␈ε(␈α∂␈ε"␈αis␈αan␈α␈y␈αpositiv␈α␈e␈αn␈α␈um␈α␈ber,
␈βε,␈↓ ↓H␈ε"w␈α␈e␈αkn␈α↓o␈α␈w␈α(by␈αthe␈αde|nition␈αof␈αRiemann-in␈α␈tegrabilit␈α␈y)␈αthat␈αthere␈αexist␈αstep␈αfunc-
␈βεW␈↓ ↓H␈ε"tions␈↓ α ␈ε(f␈↓ αA␈ε"and␈↓ βλ␈∧εWβλα⊗␈↓ βλ␈ε(f␈↓ β)␈ε"such␈αthat␈↓ ∧I␈ε(f␈↓ ∧←␈ε"(␈↓ ∧k␈ε(x␈↓ ¬∞␈ε",␈↓ ¬≡␈ε".␈αε.␈αε.␈↓ ¬N␈ε",␈↓ ¬↑␈ε(x␈↓ εβ␈ε")␈ε6␈α
∀␈↓ εG␈ε(f␈↓ ε]␈ε"(␈↓ εi␈ε(x␈↓ π␈ε",␈↓ π≤␈ε".␈αε.␈αε.␈↓ πL␈ε",␈↓ π\␈ε(x␈↓ λ␈ε")␈ε6␈α∀␈↓ λE␈∧εWλEα⊗␈↓ λE␈ε(f␈↓ λZ␈ε"(␈↓ λf␈ε(x␈↓ 		␈ε",␈↓ 	→␈ε".␈αε.␈αε.␈↓ 	I␈ε",␈↓ 	Y␈ε(x␈↓ 	}␈ε"),␈αand␈αsuch
␈βεc␈↓ ∧␈␈ε%1␈↓ ¬r␈ε+k␈↓ ε⎇␈ε%1␈↓ πp␈ε+k␈↓ λz␈ε%1␈↓ 	m␈ε+k
␈βε␈␈↓ α ␈∧ε␈α α⊗␈↓ ∧I␈∧ε␈∧Iα⊗
␈βπλ␈↓ ↓H␈ε"that␈αthe␈αdi{erence␈αof␈αthe␈α
in␈α␈tegrals␈αof␈↓ ε_␈ε(f␈↓ ε9␈ε"and␈↓ π␈∧πλπα⊗␈↓ π␈ε(f␈↓ π!␈ε"is␈αless␈αthan␈ε(␈α
∂␈ε"␈α␈.␈α⊃Since␈αEq.␈α(8)␈αh␈α↓olds
␈βπ0␈↓ ε_␈∧π0ε_α⊗
␈βπ9␈↓ ↓H␈ε"for␈↓ α␈ε(f␈↓ α!␈ε"and␈↓ αg␈∧π9αgα⊗␈↓ αg␈ε(f␈↓ α|␈ε",␈αand␈αsince
␈βπb␈↓ α␈∧πbαα⊗
␈βλ∧␈↓ β≠␈ε↓X␈↓ π⊃␈ε↓X
␈βλ∞␈↓ αa␈ε"1␈↓ εW␈ε"1
␈βλ&␈↓ βq␈ε(f␈↓ ∧ε␈ε"(␈↓ ∧∩␈ε(U␈↓ ∧:␈ε",␈↓ ∧J␈ε".␈αε.␈αε.␈↓ ∧z␈ε",␈↓ ¬
␈ε(U␈↓ ε
␈ε")␈↓ ε#␈ε6∀␈↓ πg␈ε(f␈↓ π|␈ε"(␈↓ λλ␈ε(U␈↓ λ0␈ε",␈↓ λ@␈ε".␈αε.␈αε.␈↓ λp␈ε",␈↓ 	␈ε(U␈↓ 
β␈ε")
␈βλ3␈↓ ∧+␈ε+j␈↓ ¬#␈ε+j␈↓ ¬2␈ε%+␈↓ ¬P␈ε+k␈↓ ¬`␈ε9␈␈ε%1␈↓ λ!␈ε+j␈↓ 	→␈ε+j␈↓ 	(␈ε%+␈↓ 	F␈ε+k␈↓ 	V␈ε9␈␈ε%1
␈βλ:␈↓ α←␈∧λ:α←α⊗␈↓ εU␈∧λ:εUα⊗
␈βλ@␈↓ α←␈ε(n␈↓ εU␈ε(n
␈βλO␈↓ βq␈∧λOβqα⊗
␈βλW␈↓ α␈␈ε%0␈ε9∀␈↓ β+␈ε+j␈↓ β;␈ε%<␈ε+␈α␈n␈↓ εu␈ε%0␈ε9∀␈↓ π"␈ε+j␈↓ π1␈ε%<␈ε+␈α␈n
␈βλp␈↓ π⊃␈ε↓X
␈βλz␈↓ εW␈ε"1
␈β	∩␈↓ ε#␈ε6∀␈↓ πg␈∧	∩πgα⊗␈↓ πg␈ε(f␈↓ π|␈ε"(␈↓ λλ␈ε(U␈↓ λ0␈ε",␈↓ λ@␈ε".␈αε.␈αε.␈↓ λp␈ε",␈↓ 	␈ε(U␈↓ 
β␈ε"),
␈β	∨␈↓ λ!␈ε+j␈↓ 	→␈ε+j␈↓ 	(␈ε%+␈↓ 	F␈ε+k␈↓ 	V␈ε9␈␈ε%1
␈β	&␈↓ εU␈∧	&εUα⊗
␈β	,␈↓ εU␈ε(n
␈β	C␈↓ εu␈ε%0␈ε9∀␈↓ π"␈ε+j␈↓ π1␈ε%<␈ε+␈α␈n
␈β
∂␈↓ ↓H␈ε"w␈α␈e␈αconclude␈αthat␈αEq.␈α(8)␈αis␈αtrue␈αalso␈αfor␈↓ εE␈ε(f␈↓ εZ␈ε".
␈β
_␈↓ π␈∧
_π≠∂
␈β
S␈↓ α␈ε"Theorem␈αB␈αcan␈αbe␈αapplied,␈α
for␈αexample,␈αto␈αthe␈↓ π⎇␈ε/perm␈α␈utation␈αtest␈ε"␈αof␈αSection
␈β
}␈↓ ↓H␈ε"3.3.2.␈α↔Let␈α∞(␈↓ ββ␈ε(p␈↓ β&␈ε",␈↓ β6␈ε(p␈↓ βX␈ε",␈↓ βh␈ε".␈αε.␈αε.␈↓ ∧_␈ε",␈↓ ∧(␈ε(p␈↓ ∧K␈ε")␈α∂be␈α∞an␈α␈y␈α∂perm␈α␈utation␈α∞of␈α∂the␈α∞n␈α␈um␈α␈bers␈ε6␈α∂f␈ε"1,␈αε2,␈↓ 
∧␈ε".␈αε.␈αε.␈↓ 
4␈ε",␈↓ 
D␈ε(k␈↓ 
X␈ε6g␈ε";␈α∂w␈α␈e
␈β␈↓ β⊗␈ε%1␈↓ βI␈ε%2␈↓ ∧;␈ε+k
␈β*␈↓ ↓H␈ε"w␈α␈an␈α␈t␈αto␈αsh␈α↓o␈α␈w␈αthat
␈β↓␈↓ β∪␈ε"P␈↓ β+␈ε"r␈↓ β9␈ε"(␈↓ βE␈ε(U␈↓ ∧b␈ε"<␈↓ ¬⊂␈ε(U␈↓ ε-␈ε"<␈↓ ε[␈ε6↓␈αε↓␈αε↓␈↓ π∂␈ε"<␈↓ π=␈ε(U␈↓ λQ␈ε")␈α
=␈α
1/␈↓ 	9␈ε(k␈↓ 	M␈ε"!.␈↓ 
p␈ε"(10)
␈β∞␈↓ β↑␈ε+n␈ε%+␈↓ ∧∞␈ε+p␈↓ ∧+␈ε9␈␈ε%1␈↓ ¬)␈ε+n␈ε%+␈↓ ¬Y␈ε+p␈↓ ¬v␈ε9␈␈ε%1␈↓ πV␈ε+n␈ε%+␈↓ λε␈ε+p␈↓ λ$␈ε9␈␈ε%1
␈β↔␈↓ ∧≡␈ε'1␈↓ ¬i␈ε'2␈↓ λ⊗␈ε-k
␈βY␈↓ ↓H␈ε"T␈α⎇o␈αpro␈α␈v␈α␈e␈αthis,␈αassume␈αthat␈αthe␈αsequence␈ε6␈αh␈↓ εV␈ε(U␈↓ π↓␈ε6i␈ε"␈αis␈↓ π=␈ε(k␈↓ πQ␈ε"-distributed,␈αand␈αlet
␈βe␈↓ εn␈ε+n
␈β
+␈↓ ¬F␈ε"1,␈↓ ε*␈ε"if␈↓ εL␈ε(x␈↓ πλ␈ε"<␈↓ π6␈ε(x␈↓ πq␈ε"<␈↓ λ∨␈ε6↓␈αε↓␈αε↓␈↓ λS␈ε"<␈↓ 	↓␈ε(x␈↓ 	3␈ε";
␈β
-␈↓ ¬0␈ε↓~
␈β
8␈↓ ε`␈ε+p␈↓ πJ␈ε+p␈↓ 	∃␈ε+p
␈β
A␈↓ εp␈ε'1␈↓ πZ␈ε'2␈↓ 	%␈ε-k
␈β
F␈↓ β3␈ε(f␈↓ βI␈ε"(␈↓ βU␈ε(x␈↓ βx␈ε",␈↓ ∧λ␈ε".␈αε.␈αε.␈↓ ∧8␈ε",␈↓ ∧H␈ε(x␈↓ ∧l␈ε")␈α
=
␈β
S␈↓ βi␈ε%1␈↓ ∧\␈ε+k
␈β
a␈↓ ¬F␈ε"0,␈↓ ε*␈ε"otherwise.
␈β∞2␈↓ ↓H␈ε"We␈αha␈α␈v␈α␈e
␈β∞R␈↓ β≥␈ε"P␈↓ β5␈ε"r␈↓ βD␈ε"(␈↓ βP␈ε(U␈↓ ∧m␈ε"<␈↓ ¬≠␈ε(U␈↓ ε8␈ε"<␈↓ εf␈ε6↓␈αε↓␈αε↓␈↓ π~␈ε"<␈↓ πH␈ε(U␈↓ λ[␈ε")
␈β∞↑␈↓ βh␈ε+n␈ε%+␈↓ ∧→␈ε+p␈↓ ∧6␈ε9␈␈ε%␈α␈1␈↓ ¬3␈ε+n␈ε%+␈↓ ¬d␈ε+p␈↓ ε↓␈ε9␈␈ε%␈α␈1␈↓ π`␈ε+n␈ε%+␈↓ λ⊃␈ε+p␈↓ λ/␈ε9␈␈ε%␈α␈1
␈β∞h␈↓ ∧)␈ε'1␈↓ ¬t␈ε'2␈↓ λ!␈ε-k
␈β∂β␈↓ ∧↔␈ε↓Z␈↓ ¬ε␈ε↓Z
␈β∂	␈↓ ∧;␈ε%1␈↓ ¬*␈ε%1
␈β∂/␈↓ βi␈ε"=␈↓ ∧P␈ε6↓␈αε↓␈αε↓␈↓ ¬@␈ε(f␈↓ ¬U␈ε"(␈↓ ¬a␈ε(x␈↓ ε∧␈ε",␈↓ ε∀␈ε".␈αε.␈αε.␈↓ εD␈ε",␈↓ εT␈ε(x␈↓ εy␈ε")␈ε(␈αεd␈↓ π≡␈ε(x␈↓ πG␈ε".␈αε.␈αε.␈↓ πw␈ε(d␈↓ λ␈ε(x
␈β∂;␈↓ ¬u␈ε%1␈↓ εh␈ε+k␈↓ π2␈ε%1␈↓ λ∨␈ε+k
␈β∂R␈↓ ∧+␈ε%0␈↓ ¬~␈ε%0
␈β∂g␈↓ ∧↔␈ε↓Z␈↓ ¬"␈ε↓Z␈↓ ε.␈ε↓Z␈↓ πT␈ε↓Z
␈β∂m␈↓ ∧;␈ε%1␈↓ ¬F␈ε+x␈↓ εR␈ε+x␈↓ πx␈ε+x
␈β∂v␈↓ ¬V␈ε-p␈↓ εc␈ε-p␈↓ λ	␈ε-p
␈β∂{␈↓ εq␈ε'3␈↓ λ↔␈ε'2␈↓ 	7␈ε"1
␈β∂|␈↓ ¬d␈ε-k
␈β⊂∪␈↓ βi␈ε"=␈↓ ∧V␈ε(d␈↓ ∧j␈ε(x␈↓ ¬x␈ε6↓␈αε↓␈αε↓␈↓ π
␈ε(d␈↓ π≥␈ε(x␈↓ λ0␈ε(d␈↓ λC␈ε(x␈↓ λ␈␈ε"=␈↓ 	S␈ε".
␈β⊂∨␈↓ ∧}␈ε+p␈↓ π1␈ε+p␈↓ λW␈ε+p
␈β⊂'␈↓ 	1␈∧⊂'	1α∨
␈β⊂)␈↓ ¬∞␈ε-k␈↓ πA␈ε'2␈↓ λg␈ε'1
␈β⊂-␈↓ 	1␈ε(k␈↓ 	E␈ε"!
␈β⊂6␈↓ ∧+␈ε%0␈↓ ¬6␈ε%0␈↓ εB␈ε%0␈↓ πh␈ε%0
␈β⊂k␈↓ ↓H␈ε2Corollary␈α
P.␈ε/␈α∩If␈α
a␈↓ βX␈ε"[␈↓ βh␈ε"0,␈αε1)␈ε/␈α
sequence␈α
is␈↓ ¬f␈ε(k␈↓ ¬z␈ε/-distributed,␈αit␈α
satis|es␈α
the␈α
perm␈α␈utation␈α
test
␈β⊃⊗␈↓ ↓H␈ε/of␈αorder␈↓ αP␈ε(k␈↓ αd␈ε/,␈αin␈αthe␈αsense␈αof␈αEq.␈ε"␈α(10)␈ε/.
␈β⊃∨␈↓ ε∃␈∧⊃∨ε∃≠∂
␈β∪(

␈β↓U␈↓ ↓H␈ε"148␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα$␈↓ α␈ε"We␈αcan␈αalso␈αsh␈α↓o␈α␈w␈αthat␈αthe␈↓ ¬<␈ε/serial␈αcorrelation␈αtest␈ε"␈αis␈αsatis|ed:
␈βαf␈↓ ↓H␈ε2Corollary␈α⊂S.␈ε/␈α!If␈α⊂a␈↓ βt␈ε"[␈↓ ∧∧␈ε"0,␈αε1)␈ε/␈α⊃sequence␈α⊂is␈ε"␈α⊂(␈↓ ε!␈ε(k␈↓ ε@␈ε"+␈α1)␈ε/-distributed,␈α⊃the␈α⊂serial␈α⊂correlation
␈ββ⊃␈↓ ↓H␈ε/coe}cien␈α␈t␈αbet␈α␈w␈α␈een␈↓ βy␈ε(U␈↓ ∧0␈ε/and␈↓ ∧v␈ε(U␈↓ ¬\␈ε/tends␈αto␈αzero:
␈ββ≡␈↓ ∧⊃␈ε+n␈↓ ¬∂␈ε+n␈ε%␈α␈+␈↓ ¬?␈ε+k
␈ββK␈↓ ε∩␈ε↓␈␈↓ π∀␈ε↓↓␈↓ π"␈ε↓␈␈↓ λS␈ε↓↓
␈ββO␈↓ ∧4␈ε↓P␈↓ ε@␈ε↓P␈↓ πQ␈ε↓P
␈ββe␈↓ ∧→␈ε%1␈↓ ε%␈ε%1␈↓ π6␈ε%1
␈ββj␈↓ ∧`␈ε(U␈↓ ¬λ␈ε(U␈↓ ¬f␈ε6␈␈↓ εl␈ε(U␈↓ π⎇␈ε(U
␈ββv␈↓ ∧x␈ε+j␈↓ ¬ ␈ε+j␈↓ ¬/␈ε%+␈↓ ¬M␈ε+k␈↓ π¬␈ε+j␈↓ λ⊗␈ε+j␈↓ λ%␈ε%+␈↓ λB␈ε+k
␈ββz␈↓ ∧↔␈ε+n␈↓ ε$␈ε+n␈↓ π4␈ε+n
␈ββ}␈↓ ∧↔␈∧β}∧↔α∪␈↓ ε$␈∧β}ε$α∪␈↓ π4␈∧β}π4α∪
␈β∧¬␈↓ αI␈ε"lim␈↓ 	n␈ε"=␈α
0.
␈β∧ε␈↓ β∪␈ε↓q
␈β∧∀␈↓ β7␈ε↓␈␈↓ ∧r␈ε↓␈␈↓ ¬u␈ε↓↓␈↓ ε∩␈ε↓↓␈↓ ε ␈ε↓␈␈↓ λ¬␈ε↓␈␈↓ 	5␈ε↓↓␈↓ 	R␈ε↓↓
␈β∧_␈↓ βf␈ε↓P␈↓ ¬!␈ε↓P␈↓ εO␈ε↓P␈↓ λ3␈ε↓P
␈β∧→␈↓ β∪␈∧∧→β∪αεN
␈β∧$␈↓ β7␈∧∧$β7αε*␈↓ εβ␈ε%2␈↓ 	C␈ε%2
␈β∧%␈↓ α<␈ε+n␈ε9␈α␈!1
␈β∧.␈↓ βK␈ε%1␈↓ ∧/␈ε%2␈↓ ¬ε␈ε%1␈↓ ε4␈ε%1␈↓ π_␈ε%2␈↓ λ_␈ε%1
␈β∧3␈↓ ∧∩␈ε(U␈↓ ∧F␈ε6␈␈↓ ¬M␈ε(U␈↓ ε{␈ε(U␈↓ πY␈ε6␈␈↓ λ←␈ε(U
␈β∧?␈↓ ¬f␈ε+j␈↓ λx␈ε+j␈↓ 	π␈ε%+␈↓ 	%␈ε+k
␈β∧D␈↓ βI␈ε+n␈↓ ¬∧␈ε+n␈↓ ε2␈ε+n␈↓ λ↔␈ε+n
␈β∧E␈↓ ∧*␈ε+j␈↓ π∪␈ε+j␈↓ π"␈ε%+␈↓ π@␈ε+k
␈β∧G␈↓ βI␈∧∧GβIα∪␈↓ ¬∧␈∧∧G¬∧α∪␈↓ ε2␈∧∧Gε2α∪␈↓ λ↔␈∧∧Gλ↔α∪
␈β¬⊃␈↓ ↓H␈ε"(All␈αsummations␈αhere␈αare␈αfor␈α0␈ε6␈α
∀␈↓ ¬b␈ε(j␈↓ ¬␈␈ε"<␈ε(␈α
n␈ε".)
␈β¬T␈↓ ↓H␈ε/Pro␈α↓of.␈ε"␈α→By␈αTheorem␈αB␈↓ ∧)␈ε",␈αthe␈αquan␈α␈tities
␈βε⊂␈↓ α,␈ε↓P␈↓ ∧O␈ε↓P␈↓ ε ␈ε↓P␈↓ λ~␈ε↓P␈↓ 	g␈ε↓P
␈βε%␈↓ ¬_␈ε%2␈↓ εi␈ε%2
␈βε&␈↓ α⊃␈ε%1␈↓ ∧4␈ε%1␈↓ ε¬␈ε%1␈↓ π␈␈ε%1␈↓ 	L␈ε%1
␈βε+␈↓ αX␈ε(U␈↓ β␈ε(U␈↓ βV␈ε",␈↓ ∧{␈ε(U␈↓ ¬'␈ε",␈↓ εL␈ε(U␈↓ π!␈ε",␈↓ λF␈ε(U␈↓ λn␈ε",␈↓ 
∪␈ε(U
␈βε7␈↓ αq␈ε+j␈↓ β→␈ε+j␈↓ β(␈ε%+␈↓ βE␈ε+k␈↓ λ←␈ε+j␈↓ 
+␈ε+j␈↓ 
;␈ε%+␈↓ 
X␈ε+k
␈βε<␈↓ α∂␈ε+n␈↓ ∧2␈ε+n␈↓ ¬∪␈ε+j␈↓ εβ␈ε+n␈↓ εd␈ε+j␈↓ εs␈ε%+␈↓ π⊃␈ε+k␈↓ π⎇␈ε+n␈↓ 	J␈ε+n
␈βε?␈↓ α∂␈∧ε?α∂α∪␈↓ ∧2␈∧ε?∧2α∪␈↓ εβ␈∧ε?εβα∪␈↓ π⎇␈∧ε?π⎇α∪␈↓ 	J␈∧ε?	Jα∪
␈βε⎇␈↓ ¬∩␈ε%1␈↓ ¬?␈ε%1␈↓ ¬l␈ε%1␈↓ ε→␈ε%1␈↓ εG␈ε%1
␈βπα␈↓ ↓H␈ε"tend␈αto␈αthe␈αrespectiv␈α␈e␈αlimits␈↓ ¬%␈ε",␈↓ ¬R␈ε",␈↓ ¬␈␈ε",␈↓ ε,␈ε",␈↓ εf␈ε"as␈ε(␈αn␈ε6␈α	!␈α
1␈ε".
␈βπ␈↓ λ5␈∧πλ5≠∂
␈βπ∪␈↓ ¬∩␈ε%4␈↓ ¬?␈ε%3␈↓ ¬l␈ε%3␈↓ ε→␈ε%2␈↓ εG␈ε%2
␈βπ⊗␈↓ ¬∩␈∧π⊗¬∩α⊂␈↓ ¬?␈∧π⊗¬?α⊂␈↓ ¬l␈∧π⊗¬lα⊂␈↓ ε→␈∧π⊗ε→α⊂␈↓ εG␈∧π⊗εGα⊂
␈βπD␈↓ α␈ε"Let␈α∂us␈α∂n␈α↓o␈α␈w␈α∂consider␈α⊂some␈α∂sligh␈α␈tly␈α∂m␈α↓ore␈α∂general␈α∂distribution␈α∂properties␈α∂of
␈βπo␈↓ ↓H␈ε"sequences.␈α→We␈α∂ha␈α␈v␈α␈e␈α⊂de|ned␈α∂the␈α∂n␈α↓otion␈α∂of␈↓ ε|␈ε(k␈↓ π⊃␈ε"-distribution␈α∂by␈α∂considering␈α∂all␈α∂of
␈βλ~␈↓ ↓H␈ε"the␈αadjacen␈α␈t␈↓ β⊗␈ε(k␈↓ β*␈ε"-tuples;␈αfor␈αexample,␈αa␈αsequence␈αis␈α2-distributed␈αif␈αand␈αonly␈αif␈αthe
␈βλF␈↓ ↓H␈ε"poin␈α␈ts
␈βλq␈↓ β=␈ε"(␈↓ βI␈ε(U␈↓ βq␈ε",␈↓ ∧↓␈ε(U␈↓ ∧)␈ε"),␈α∞(␈↓ ∧Y␈ε(U␈↓ ¬↓␈ε",␈↓ ¬⊃␈ε(U␈↓ ¬9␈ε"),␈α∞(␈↓ ¬i␈ε(U␈↓ ε⊃␈ε",␈↓ ε!␈ε(U␈↓ εI␈ε"),␈α∞(␈↓ εy␈ε(U␈↓ π!␈ε",␈↓ π1␈ε(U␈↓ πY␈ε"),␈α∞(␈↓ λ	␈ε(U␈↓ λ1␈ε",␈↓ λA␈ε(U␈↓ λi␈ε"),␈↓ 	
␈ε".␈αε.␈αε.
␈βλ}␈↓ βb␈ε%0␈↓ ∧~␈ε%1␈↓ ∧r␈ε%1␈↓ ¬*␈ε%2␈↓ εα␈ε%2␈↓ ε:␈ε%3␈↓ π∩␈ε%3␈↓ πJ␈ε%4␈↓ λ"␈ε%4␈↓ λZ␈ε%5
␈β	6␈↓ ↓H␈ε"are␈α∂equidistributed␈α∂in␈α∂the␈α∂unit␈α∂square.␈α~It␈α∂is␈α∂quite␈α∂possible,␈α⊂h␈α↓o␈α␈w␈α␈ev␈α␈er,␈α⊂that␈α∂this
␈β	a␈↓ ↓H␈ε"can␈α
happen␈αwhile␈αalternate␈α
pairs␈αof␈αpoin␈α␈ts␈α
(␈↓ εm␈ε(U␈↓ π∃␈ε",␈↓ π%␈ε(U␈↓ πM␈ε"),␈α(␈↓ πz␈ε(U␈↓ λ"␈ε",␈↓ λ2␈ε(U␈↓ λZ␈ε"),␈α(␈↓ 	π␈ε(U␈↓ 	/␈ε",␈↓ 	?␈ε(U␈↓ 	g␈ε"),␈↓ 
λ␈ε".␈αε.␈αε.␈↓ 
=␈ε"are␈ε/␈α
n␈α↓ot
␈β	n␈↓ πε␈ε%1␈↓ π>␈ε%2␈↓ λ∪␈ε%3␈↓ λK␈ε%4␈↓ 	 ␈ε%5␈↓ 	X␈ε%6
␈β

␈↓ ↓H␈ε"equidistributed;␈α	if␈α	the␈α	densit␈α␈y␈α	of␈α	poin␈α␈ts␈αλ(␈↓ ε4␈ε(U␈↓ π≤␈ε",␈↓ π,␈ε(U␈↓ πf␈ε")␈α	is␈α	de|cien␈α␈t␈α	in␈αλsome␈α	area,␈α	the
␈β
→␈↓ εM␈ε%2␈ε+n␈ε9␈␈ε%␈α␈1␈↓ πD␈ε%2␈ε+␈α↓n
␈β
8␈↓ ↓H␈ε"other␈α
poin␈α␈ts␈α
(␈↓ β∨␈ε(U␈↓ βY␈ε",␈↓ βi␈ε(U␈↓ ∧P␈ε")␈α
migh␈α␈t␈α
compensate.␈α∪F␈α⎇or␈α
example,␈α∞the␈αperiodic␈α
binary
␈β
D␈↓ β7␈ε%2␈ε+␈α↓n␈↓ ∧α␈ε%2␈ε+n␈ε%+␈α␈1
␈β
c␈↓ ↓H␈ε"sequence
␈β:␈↓ α\␈ε6h␈↓ αh␈ε(X␈↓ β_␈ε6i␈ε"␈α
=␈α
0,␈αε0,␈αε0,␈αε1,␈α_0,␈αε0,␈αε0,␈αε1,␈α_1,␈αε1,␈αε0,␈αε1,␈α_1,␈αε1,␈αε0,␈αε1,␈α_0,␈αε0,␈αε0,␈αε1,␈↓ 	↑␈ε".␈αε.␈αε.␈↓ 
∞␈ε",␈↓ 
p␈ε"(11)
␈βF␈↓ βε␈ε+n
␈β⊃␈↓ ↓H␈ε"with␈αa␈αperiod␈αof␈αlength␈α16,␈αis␈αseen␈αto␈αbe␈α3-distributed;␈αy␈α␈et␈αthe␈αsequence␈αof␈αev␈α␈en-
␈β<␈↓ ↓H␈ε"n␈α␈um␈α␈bered␈α∞elemen␈α␈ts␈ε6␈α∞h␈↓ ∧⊃␈ε(X␈↓ ∧P␈ε6i␈ε"␈α∞=␈α
0,␈α∞0,␈α∂0,␈α∞0,␈α∂1,␈α∞0,␈α∂1,␈α∞0,␈↓ πo␈ε".␈αε.␈αε.␈↓ λ'␈ε"has␈α∞three␈α∞times␈α
as␈α∞man␈α␈y
␈βI␈↓ ∧/␈ε%2␈ε+n
␈βg␈↓ ↓H␈ε"zeros␈αas␈α
ones,␈αwhile␈αthe␈αsubsequence␈αof␈αodd-n␈α␈um␈α␈bered␈αelemen␈α␈ts␈ε6␈αh␈↓ 	9␈ε(X␈↓ 
%␈ε6i␈ε"␈α
=␈α
0,␈α1,
␈βt␈↓ 	W␈ε%2␈ε+n␈ε%+␈α␈1
␈β
∪␈↓ ↓H␈ε"0,␈α1,␈α1,␈α1,␈α1,␈α1,␈↓ β8␈ε".␈αε.␈αε.␈↓ βn␈ε"has␈αthree␈αtimes␈αas␈αman␈α␈y␈αones␈αas␈αzeros.
␈β
>␈↓ α␈ε"If␈αa␈αsequence␈ε6␈αh␈↓ βn␈ε(U␈↓ ∧→␈ε6i␈ε"␈αis␈ε6␈α1␈ε"-distributed,␈αexample␈α(11)␈αsh␈α↓o␈α␈ws␈αthat␈αit␈αis␈αn␈α↓ot␈αat␈αall
␈β
J␈↓ ∧ε␈ε+n
␈β
i␈↓ ↓H␈ε"obvious␈αthat␈αthe␈αsubsequence␈αof␈αalternate␈αterms␈ε6␈αh␈↓ πL␈ε(U␈↓ λπ␈ε6i␈ε"␈α
=␈↓ λK␈ε(U␈↓ λs␈ε",␈↓ 	λ␈ε(U␈↓ 	0␈ε",␈↓ 	E␈ε(U␈↓ 	m␈ε",␈↓ 
α␈ε(U␈↓ 
*␈ε",␈↓ 
?␈ε".␈αε.␈αε.␈↓ 
t␈ε"will
␈β
v␈↓ πe␈ε%2␈ε+n␈↓ λc␈ε%0␈↓ 	!␈ε%2␈↓ 	↑␈ε%4␈↓ 
≠␈ε%6
␈β∞∀␈↓ ↓H␈ε"be␈ε6␈α
1␈ε"-distributed␈α∞or␈α
ev␈α␈en␈α∞1-distributed.␈α∀But␈α∞w␈α␈e␈α
shall␈α∞see␈α
that␈ε6␈α∞h␈↓ 	7␈ε(U␈↓ 	q␈ε6i␈ε"␈α∞is,␈α∞in␈α
fact,
␈β∞!␈↓ 	P␈ε%2␈ε+n
␈β∞@␈↓ ↓H␈ε61␈ε"-distributed,␈αand␈αm␈α␈uch␈αm␈α↓ore␈αis␈αtrue.
␈β∂α␈↓ ↓H␈ε2De|nition␈αE.␈ε/␈α→A␈↓ βQ␈ε"[␈↓ βa␈ε"0,␈αε1)␈ε/␈αsequence␈ε6␈αh␈↓ ¬M␈ε(U␈↓ ¬y␈ε6i␈ε/␈αis␈αsaid␈αto␈αbe␈ε"␈α(␈ε(m␈ε"␈α␈,␈↓ λ⊗␈ε(k␈↓ λ*␈ε")␈ε/-distributed␈αif
␈β∂∞␈↓ ¬f␈ε+n
␈β∂U␈↓ ↓l␈ε"P␈↓ α∧␈ε"r␈↓ α∩␈ε"(␈↓ α≡␈ε(u␈↓ αL␈ε6∀␈↓ αz␈ε(U␈↓ βw␈ε"<␈↓ ∧%␈ε(v␈↓ ∧E␈ε",␈↓ ∧[␈ε(u␈↓ ¬	␈ε6∀␈↓ ¬7␈ε(U␈↓ εa␈ε"<␈↓ π∂␈ε(v␈↓ π0␈ε",␈↓ πF␈ε".␈αε.␈αε.␈↓ πv␈ε",␈↓ λ␈ε(u␈↓ λ;␈ε6∀␈↓ λi␈ε(U␈↓ 
A␈ε"<␈↓ 
o␈ε(v␈↓ ⊃␈ε")
␈β∂a␈↓ α3␈ε%1␈↓ β∪␈ε+m␈α␈n␈ε%+␈↓ β]␈ε+j␈↓ ∧6␈ε%1␈↓ ∧p␈ε%2␈↓ ¬P␈ε+mn␈ε%␈α␈+␈↓ ε≠␈ε+j␈↓ ε*␈ε%+1␈↓ π ␈ε%2␈↓ λ ␈ε+k␈↓ 	α␈ε+m␈α␈n␈ε%+␈↓ 	L␈ε+j␈↓ 	\␈ε%+␈↓ 	y␈ε+k␈↓ 

␈ε9␈␈ε%␈α␈1␈↓ ␈ε+k
␈β⊂⊗␈↓ λ␈ε"=␈α
(␈↓ λ:␈ε(v␈↓ λc␈ε6␈␈↓ 	∂␈ε(u␈↓ 	3␈ε")␈↓ 	E␈ε".␈αε.␈αε.␈↓ 	u␈ε"(␈↓ 
↓␈ε(v␈↓ 
+␈ε6␈␈↓ 
W␈ε(u␈↓ 
|␈ε")
␈β⊂"␈↓ λL␈ε%1␈↓ 	$␈ε%1␈↓ 
∪␈ε+k␈↓ 
l␈ε+k
␈β⊂m␈↓ ↓H␈ε/for␈αall␈αch␈α↓oices␈αof␈αreal␈αn␈α␈um␈α␈bers␈↓ ¬-␈ε(u␈↓ ¬O␈ε/,␈↓ ¬f␈ε(v␈↓ ε⊃␈ε/with␈ε"␈α0␈ε6␈α
∀␈↓ π-␈ε(u␈↓ πZ␈ε"<␈↓ λλ␈ε(v␈↓ λ1␈ε6∀␈ε"␈α
1␈ε/␈αfor␈ε"␈α1␈ε6␈α
∀␈↓ 
␈ε(r␈↓ 
~␈ε6∀␈↓ 
H␈ε(k␈↓ 
\␈ε/,␈αand
␈β⊂y␈↓ ¬B␈ε+r␈↓ ¬w␈ε+r␈↓ πB␈ε+r␈↓ λ→␈ε+r
␈β⊃_␈↓ ↓H␈ε/for␈αall␈αin␈α␈tegers␈↓ β7␈ε(j␈↓ βV␈ε/with␈ε"␈α0␈ε6␈α
∀␈↓ ∧r␈ε(j␈↓ ¬∂␈ε"<␈ε(␈α
m␈ε/.
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"149
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα$␈↓ ↓H␈ε"Th␈α␈us␈α
a␈↓ α=␈ε(k␈↓ αQ␈ε"-distributed␈αsequence␈α
is␈α
the␈α
special␈αcase␈ε(␈α
m␈ε"␈α
=␈α
1␈α
in␈α
De|nition␈α
E␈↓ 
≠␈ε";␈αthe␈α
case
␈βαO␈↓ ↓H␈ε(m␈ε"␈α=␈α
2␈α∞means␈α∞that␈α∞the␈↓ ∧D␈ε(k␈↓ ∧X␈ε"-tuples␈α∞starting␈α∞in␈α∞ev␈α␈en␈α
positions␈α∞m␈α␈ust␈α∞ha␈α␈v␈α␈e␈α∞the␈α
same
␈βαz␈↓ ↓H␈ε"densit␈α␈y␈αas␈αthe␈↓ β.␈ε(k␈↓ βB␈ε"-tuples␈αstarting␈αin␈αodd␈αpositions,␈αetc.
␈ββ&␈↓ α␈ε"Sev␈α␈eral␈αproperties␈αof␈αDe|nition␈αE␈αare␈αobvious:
␈ββv␈↓ ↓Y␈ε/An␈ε"␈α(␈ε(m␈ε",␈↓ αO␈ε(k␈↓ αc␈ε")␈ε/-distributed␈αsequence␈αis␈ε"␈α(␈ε(m␈ε",␈ε(␈αε∀␈ε")␈ε/-distributed␈αfor␈ε"␈α1␈ε6␈α
∀␈ε(␈α
∀␈ε6␈α
∀␈↓ 	[␈ε(k␈↓ 	o␈ε/.␈↓ 
p␈ε"(12)
␈β∧;␈↓ ↓Y␈ε/An␈ε"␈α(␈ε(m␈ε",␈↓ αO␈ε(k␈↓ αc␈ε")␈ε/-distributed␈αsequence␈αis␈ε"␈α(␈ε(d␈ε"␈α↓,␈↓ ε≠␈ε(k␈↓ ε/␈ε")␈ε/-distributed␈αfor␈αall␈αdivisors␈ε(␈αd␈ε/␈αof␈ε(␈αm␈ε/.␈↓ 
p␈ε"(13)
␈β¬⊃␈↓ ↓H␈ε"We␈α⊂can␈α⊂also␈α⊂de|ne␈α⊂the␈α⊂concept␈α⊂of␈α⊂an␈α⊂(␈ε(m␈ε",␈↓ εt␈ε(k␈↓ πλ␈ε")-distributed␈ε(␈α⊂b␈ε"-ary␈α⊂sequence,␈α⊃as␈α⊂in
␈β¬<␈↓ ↓H␈ε"De|nition␈α∞D␈↓ β∞␈ε";␈α∂and␈α∞the␈α∞pro␈α↓of␈α∞of␈α∞Theorem␈α∞A␈α∞remains␈α∞v␈α⎇alid␈α∞for␈α∂(␈ε(m␈ε"␈α␈,␈↓ 	T␈ε(k␈↓ 	h␈ε")-distributed
␈β¬g␈↓ ↓H␈ε"sequences.
␈βε∩␈↓ α␈ε"The␈αnext␈αtheorem,␈αwhich␈αis␈αin␈αman␈α␈y␈αw␈α␈a␈α␈ys␈αrather␈αsurprising,␈αsh␈α↓o␈α␈ws␈αthat␈αthe
␈βε>␈↓ ↓H␈ε"propert␈α␈y␈α∞of␈α∞being␈ε6␈α∂1␈ε"-distributed␈α∞is␈α∞v␈α␈ery␈α∞strong␈α∂indeed,␈α∂m␈α␈uch␈α∞stronger␈α∞than␈α∞w␈α␈e
␈βεi␈↓ ↓H␈ε"imagined␈αit␈αto␈αbe␈αwhen␈αw␈α␈e␈α|rst␈αconsidered␈αthe␈αde|nition␈αof␈αthe␈αconcept.
␈βπ)␈↓ ↓H␈ε2Theorem␈α∞C␈ε"␈α∞(Iv␈α⎇an␈↓ βf␈ε"Niv␈α␈en␈α∞and␈α∞H.␈α∞S.␈↓ ¬v␈ε"Zuck␈α␈erman)␈ε2.␈ε/␈α≥An␈ε6␈α∞1␈ε/-distributed␈α∞sequence␈α∞is
␈βπU␈↓ ↓H␈ε"(␈ε(m␈ε"␈α␈,␈↓ αβ␈ε(k␈↓ α_␈ε")␈ε/-distributed␈αfor␈αall␈αpositiv␈α␈e␈αin␈α␈tegers␈ε(␈αm␈ε/␈αand␈↓ πL␈ε(k␈↓ π`␈ε/.
␈βλ∃␈↓ ↓H␈ε/Pro␈α↓of.␈ε"␈α≥It␈α∂su}ces␈α∞to␈α∂pro␈α␈v␈α␈e␈α∞the␈α∂theorem␈α∞for␈ε(␈α∂b␈ε"-ary␈α∂sequences,␈α∂by␈α∞using␈α∂the␈α∞gen-
␈βλ@␈↓ ↓H␈ε"eralization␈α∂of␈α⊂Theorem␈α∂A␈α⊂just␈α∂men␈α␈tioned.␈α≠F␈α⎇urtherm␈α↓ore,␈α⊂w␈α␈e␈α∂ma␈α␈y␈α⊂assume␈α∂that
␈βλl␈↓ ↓H␈ε(m␈ε"␈α	=␈↓ α∨␈ε(k␈↓ α4␈ε",␈αbecause␈α(12)␈αand␈α(13)␈αtell␈αus␈αthat␈αthe␈αsequence␈αwill␈αbe␈α(␈ε(m␈ε"␈α␈,␈↓ 	T␈ε(k␈↓ 	h␈ε")-distributed
␈β	↔␈↓ ↓H␈ε"if␈αit␈αis␈α(␈ε(m␈↓ α]␈ε(k␈↓ αr␈ε",␈ε(␈αεm␈↓ β!␈ε(k␈↓ β5␈ε")-distributed.
␈β	B␈↓ α␈ε"So␈α	w␈α␈e␈α	will␈αλpro␈α␈v␈α␈e␈α	that␈ε/␈α	an␈α␈y␈ε6␈α	1␈ε/-distributed␈ε(␈αλb␈ε/␈α↓-ary␈αλsequence␈↓ λd␈ε(X␈↓ 	⊃␈ε/,␈↓ 	%␈ε(X␈↓ 	R␈ε/,␈↓ 	e␈ε".␈αε.␈αε.␈↓ 
_␈ε/is␈ε"␈α	(␈ε(m␈ε",␈ε(␈αεm␈ε"␈α␈)␈ε/-
␈β	N␈↓ 	α␈ε%0␈↓ 	C␈ε%1
␈β	m␈↓ ↓H␈ε/distributed␈α∞for␈α∞all␈α
positiv␈α␈e␈α∞in␈α␈tegers␈ε(␈α∞m␈ε/.␈ε"␈α⊗Our␈α∞pro␈α↓of␈α∞is␈α∞a␈α∞simpli|ed␈α∞v␈α␈ersion␈α∞of␈α
the
␈β
_␈↓ ↓H␈ε"original␈α	one␈α	giv␈α␈en␈α	by␈α	Niv␈α␈en␈α	and␈α	Zuck␈α␈erman␈α	in␈ε/␈α	Paci|c␈α	J.␈α
Math.␈ε2␈α	1␈ε"␈α	(1951),␈↓ 
$␈ε"103↑109␈↓ "␈ε".
␈β
D␈↓ α␈ε"The␈α∞k␈α␈ey␈α∞idea␈α∞w␈α␈e␈α∞shall␈α
use␈α∞is␈α∞an␈α∞importan␈α␈t␈α∞technique␈α∞that␈α∞applies␈α∞to␈α
man␈α␈y
␈β
o␈↓ ↓H␈ε"situations␈α⊂in␈α⊂mathematics:␈α∀\If␈α⊂the␈α⊃sum␈α⊂of␈ε(␈α⊂m␈ε"␈α⊂quan␈α␈tities␈α⊂and␈α⊂the␈α⊂sum␈α⊂of␈α⊂their
␈β~␈↓ ↓H␈ε"squares␈αare␈αboth␈αconsisten␈α␈t␈αwith␈αthe␈αh␈α␈ypothesis␈αthat␈αthe␈ε(␈αm␈ε"␈αquan␈α␈tities␈αare␈αequal,
␈βE␈↓ ↓H␈ε"then␈αthat␈α
h␈α␈ypothesis␈αis␈α
true."␈α⊃In␈α
a␈α
strong␈αform,␈α
this␈αprinciple␈α
ma␈α␈y␈αbe␈α
stated␈αas
␈βp␈↓ ↓H␈ε"follo␈α␈ws:
␈β1␈↓ ↓H␈ε2Lemma␈αE.␈ε/␈α_Giv␈α␈en␈ε(␈αm␈ε/␈αsequences␈αof␈αn␈α␈um␈α␈bers␈ε6␈αh␈↓ ε}␈ε(y␈↓ π1␈ε6i␈ε"␈α
=␈↓ πu␈ε(y␈↓ λ%␈ε/,␈↓ λ;␈ε(y␈↓ λk␈ε/,␈↓ 	␈ε".␈αε.␈αε.␈↓ 	6␈ε/for␈ε"␈α1␈ε6␈α
∀␈↓ 
8␈ε(j␈↓ 
U␈ε6∀␈ε(␈α
m␈ε/␈α␈,
␈β>␈↓ π∂␈ε+j␈↓ π≡␈ε+n␈↓ λε␈ε+j␈↓ λ⊗␈ε%0␈↓ λL␈ε+j␈↓ λ[␈ε%1
␈β\␈↓ ↓H␈ε/suppose␈αthat
␈β
α␈↓ ∧2␈ε"lim␈↓ ∧q␈ε"(␈↓ ∧⎇␈ε(y␈↓ ¬9␈ε"+␈↓ ¬e␈ε(y␈↓ ε ␈ε"+␈↓ εL␈ε6↓␈αε↓␈αε↓␈↓ ε}␈ε"+␈↓ π*␈ε(y␈↓ πi␈ε")␈↓ π␈␈ε"=␈ε(␈α
m␈ε",
␈β
∞␈↓ ¬∂␈ε%1␈ε+n␈↓ ¬v␈ε%2␈ε+n␈↓ π<␈ε+m␈α␈n
␈β
"␈↓ ∧$␈ε+n␈ε9␈α␈!1
␈β
3␈↓ 
p␈ε"(14)
␈β
A␈↓ ¬␈ε%2␈↓ ¬u␈ε%2␈↓ π<␈ε%2␈↓ λb␈ε%2
␈β
I␈↓ βy␈ε"lim␈αεsup␈↓ ∧m␈ε"(␈↓ ∧y␈ε(y␈↓ ¬6␈ε"+␈↓ ¬b␈ε(y␈↓ ε∨␈ε"+␈↓ εK␈ε6↓␈αε↓␈αε↓␈↓ ε⎇␈ε"+␈↓ π)␈ε(y␈↓ πi␈ε")␈↓ π␈␈ε6∀␈ε(␈α
m␈↓ λM␈ε(␈↓ λq␈ε".
␈β
Y␈↓ ¬␈ε%1␈ε+n␈↓ ¬u␈ε%2␈ε+n␈↓ π<␈ε+m␈α␈n
␈β
p␈↓ ∧	␈ε+n␈ε9!1
␈β∞"␈↓ ↓H␈ε/Then␈αfor␈αeach␈↓ β0␈ε(j␈↓ βC␈ε/,␈↓ βY␈ε"lim␈↓ ∧↑␈ε(y␈↓ ¬≡␈ε/exists␈αand␈αequals␈ε(␈α␈ε/.
␈β∞/␈↓ ∧␈ε+n␈ε9␈α␈!1␈↓ ∧p␈ε+j␈↓ ∧␈␈ε+n
␈β∞c␈↓ ↓H␈ε"An␈αincredibly␈αsimple␈αpro␈α↓of␈αof␈αthis␈αlemma␈αis␈αgiv␈α␈en␈αin␈αexercise␈α9.
␈β∞l␈↓ 	Q␈∧∞l	Q≠∂
␈β∂$␈↓ α␈ε"Resuming␈α
our␈αpro␈α↓of␈α
of␈αTheorem␈αC␈↓ ε1␈ε",␈α
let␈ε(␈α
x␈ε"␈α
=␈↓ πJ␈ε(x␈↓ πm␈ε(x␈↓ λ↔␈ε".␈αε.␈αε.␈↓ λG␈ε(x␈↓ 	α␈ε"be␈αa␈ε(␈α
b␈ε"-ary␈αn␈α␈um␈α␈ber,
␈β∂0␈↓ π↑␈ε%1␈↓ λ↓␈ε%2␈↓ λ[␈ε+m
␈β∂O␈↓ ↓H␈ε"and␈α	sa␈α␈y␈α	that␈ε(␈α	x␈ε/␈α	occurs␈ε"␈α	at␈α	position␈ε(␈α	p␈ε"␈α	of␈αλthe␈α	sequence␈α	if␈↓ πt␈ε(X␈↓ 	π␈ε(X␈↓ 
 ␈ε".␈αε.␈αε.␈↓ 
P␈ε(X␈↓ λ␈ε"=
␈β∂[␈↓ λ∩␈ε+p␈ε9␈␈ε+m␈ε%+␈α␈1␈↓ 	%␈ε+p␈ε9␈␈ε+m␈ε%+␈α␈2␈↓ 
n␈ε+p
␈β∂z␈↓ ↓H␈ε(x␈ε".␈α∀Let␈↓ α=␈ε(↔␈↓ α←␈ε"(␈ε(n␈ε")␈α∞be␈α
the␈α
n␈α␈um␈α␈ber␈α∞of␈α
occurrences␈α∞of␈ε(␈α
x␈ε"␈α∞at␈α
position␈ε(␈α∞p␈ε"␈α
when␈ε(␈α
p␈ε"␈α
<␈ε(␈αn␈ε"␈α
and
␈β⊂ε␈↓ αP␈ε+j
␈β⊂%␈↓ ↓H␈ε(p␈↓ ↓a␈ε"mod␈↓ α+␈ε(m␈ε"␈α	=␈↓ βα␈ε(j␈↓ β∃␈ε".␈α⊂Let␈↓ βq␈ε(y␈↓ ∧/␈ε"=␈↓ ∧]␈ε(↔␈↓ ∧␈␈ε"(␈ε(n␈ε"␈α␈)/␈ε(n␈ε";␈αw␈α␈e␈αwish␈αto␈αpro␈α␈v␈α␈e␈αthat
␈β⊂2␈↓ ∧β␈ε+j␈↓ ∧∩␈ε+n␈↓ ∧p␈ε+j
␈β⊂s␈↓ π1␈ε+m
␈β⊂{␈↓ ¬,␈ε"lim␈↓ ¬r␈ε(y␈↓ ε/␈ε"=␈α
1/␈ε(m␈↓ π!␈ε(b␈↓ πL␈ε".␈↓ 
p␈ε"(15)
␈β⊃λ␈↓ εβ␈ε+j␈↓ ε∪␈ε+n
␈β⊃≠␈↓ ¬≡␈ε+n␈ε9!␈α␈1
␈β∪(

␈β↓U␈↓ ↓H␈ε"150␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα$␈↓ α␈ε"First␈αw␈α␈e␈αkn␈α↓o␈α␈w␈αthat
␈βαz␈↓ λg␈ε+m
␈ββ↓␈↓ βv␈ε"lim␈↓ ∧6␈ε"(␈↓ ∧B␈ε(y␈↓ ∧⎇␈ε"+␈↓ ¬)␈ε(y␈↓ ¬e␈ε"+␈↓ ε⊃␈ε6↓␈αε↓␈αε↓␈↓ εC␈ε"+␈↓ εo␈ε(y␈↓ πo␈ε")␈α
=␈α
1/␈↓ λW␈ε(b␈↓ 	α␈ε",␈↓ 
p␈ε"(16)
␈ββ∞␈↓ ∧T␈ε%0␈ε+n␈↓ ¬;␈ε%1␈ε+n␈↓ π␈ε%(␈ε+␈α↓m␈ε9␈α␈␈␈ε%1)␈ε+n
␈ββ"␈↓ βh␈ε+n␈ε9!␈α␈1
␈ββo␈↓ ↓H␈ε"since␈α
the␈αsequence␈α
is␈ε(␈αm␈ε"␈α␈-distributed.␈α⊂By␈αLemma␈α
E␈αand␈α
Eq.␈α(16),␈α
the␈αtheorem␈α
will
␈β∧~␈↓ ↓H␈ε"be␈αpro␈α␈v␈α␈ed␈αif␈αw␈α␈e␈αcan␈αsh␈α↓o␈α␈w␈αthat
␈β∧p␈↓ ∧O␈ε%2␈↓ ¬7␈ε%2␈↓ ε}␈ε%2␈↓ 	¬␈ε%2␈ε+m
␈β∧x␈↓ β<␈ε"lim␈αεsup␈↓ ∧0␈ε"(␈↓ ∧<␈ε(y␈↓ ∧x␈ε"+␈↓ ¬$␈ε(y␈↓ ¬a␈ε"+␈↓ ε
␈ε6↓␈αε↓␈αε↓␈↓ ε?␈ε"+␈↓ εk␈ε(y␈↓ πm␈ε")␈ε6␈α
∀␈ε"␈α
1/␈ε(m␈↓ λt␈ε(b␈↓ 	.␈ε".␈↓ 
p␈ε"(17)
␈β¬λ␈↓ ∧O␈ε%0␈ε+n␈↓ ¬7␈ε%1␈ε+n␈↓ ε}␈ε%(␈ε+m␈ε9␈␈ε%␈α␈1␈α↓)␈ε+n
␈β¬∨␈↓ βL␈ε+n␈ε9␈α␈!1
␈β¬m␈↓ α␈ε"This␈α∪inequalit␈α␈y␈α∩is␈α∪n␈α↓ot␈α∩obvious␈α∪y␈α␈et;␈α∃some␈α∪rather␈α∩delicate␈α∪maneuv␈α␈ering␈α∩is
␈βε_␈↓ ↓H␈ε"necessary␈αbefore␈αw␈α␈e␈αcan␈αpro␈α␈v␈α␈e␈αit.␈α⊂Let␈↓ ε∨␈ε(q␈↓ ε>␈ε"be␈αa␈αm␈α␈ultiple␈αof␈ε(␈αm␈ε",␈αand␈αconsider
␈βεg␈↓ ¬6␈ε↓X
␈βεp␈↓ ε⊂␈ε↓∩␈↓ λ@␈ε↓∪
␈βεq␈↓ ε&␈ε(↔␈↓ εH␈ε"(␈ε(n␈ε")␈ε6␈αλ␈␈↓ π*␈ε(↔␈↓ πL␈ε"(␈ε(n␈ε6␈απ␈␈↓ λ!␈ε(q␈↓ λ4␈ε")
␈βε⎇␈↓ ε9␈ε+j␈↓ π<␈ε+j
␈βπ	␈↓ ∧∀␈ε(C␈↓ ∧0␈ε"(␈ε(n␈ε")␈α
=␈↓ λV␈ε".␈↓ 
p␈ε"(18)
␈βπ#␈↓ π*␈ε"2
␈βπ9␈↓ ¬⊗␈ε%0␈ε9∀␈↓ ¬C␈ε+j␈↓ ¬R␈ε%<␈ε+␈α␈m
␈βλ
␈↓ ↓H␈ε"This␈αis␈αthe␈αn␈α␈um␈α␈ber␈αof␈αpairs␈αof␈αoccurrences␈αof␈ε(␈αx␈ε"␈αin␈αpositions␈↓ λq␈ε(p␈↓ 	∪␈ε",␈↓ 	#␈ε(p␈↓ 	Q␈ε"with␈ε(␈αn␈ε6␈απ␈␈↓ 
k␈ε(q␈↓ λ␈ε6∀
␈βλ⊗␈↓ 	∧␈ε%1␈↓ 	6␈ε%2
␈βλ5␈↓ ↓H␈ε(p␈↓ ↓t␈ε"<␈↓ α"␈ε(p␈↓ αN␈ε"<␈ε(␈α
n␈ε"␈αand␈αwith␈↓ ∧6␈ε(p␈↓ ∧`␈ε6␈␈↓ ¬␈ε(p␈↓ ¬:␈ε"a␈αm␈α␈ultiple␈αof␈ε(␈αm␈ε".␈α⊂Consider␈αn␈α↓o␈α␈w␈αthe␈αsum
␈βλA␈↓ ↓[␈ε%1␈↓ α5␈ε%2␈↓ ∧I␈ε%2␈↓ ¬∨␈ε%1
␈βλp␈↓ ε(␈ε↓X
␈β	∩␈↓ ¬¬␈ε(S␈↓ ¬@␈ε"=␈↓ π≠␈ε(C␈↓ π7␈ε"(␈ε(n␈ε").␈↓ 
p␈ε"(19)
␈β	∨␈↓ ¬≠␈ε+N
␈β	C␈↓ ¬n␈ε%1␈ε9␈α↓∀␈ε+␈α␈n␈ε9∀␈↓ εK␈ε+N␈↓ εg␈ε%+␈↓ π¬␈ε+q
␈β
∪␈↓ ↓H␈ε"Each␈αpair␈α
of␈αoccurrences␈α
of␈ε(␈αx␈ε"␈α
in␈α
positions␈↓ εd␈ε(p␈↓ πε␈ε",␈↓ π⊗␈ε(p␈↓ πE␈ε"with␈↓ λ_␈ε(p␈↓ λE␈ε"<␈↓ λt␈ε(p␈↓ 	!␈ε"<␈↓ 	P␈ε(p␈↓ 	z␈ε"+␈↓ 
'␈ε(q␈↓ 
:␈ε",␈αwhere
␈β
∨␈↓ εw␈ε%1␈↓ π)␈ε%2␈↓ λ+␈ε%1␈↓ 	π␈ε%2␈↓ 	c␈ε%1
␈β
>␈↓ ↓H␈ε(p␈↓ ↓o␈ε6␈␈↓ α_␈ε(p␈↓ αE␈ε"is␈α
a␈αm␈α␈ultiple␈α
of␈ε(␈αm␈ε"␈α
and␈↓ ¬'␈ε(p␈↓ ¬S␈ε6∀␈↓ ε↓␈ε(N␈↓ ε$␈ε",␈αis␈α
coun␈α␈ted␈↓ πa␈ε(p␈↓ λλ␈ε"+␈↓ λ1␈ε(q␈↓ λI␈ε6␈␈↓ λr␈ε(p␈↓ 	∨␈ε"times␈α
in␈αthe␈α
total
␈β
K␈↓ ↓[␈ε%2␈↓ α+␈ε%1␈↓ ¬:␈ε%1␈↓ πt␈ε%1␈↓ 	¬␈ε%2
␈β
i␈↓ ↓H␈ε(S␈↓ απ␈ε"(namely,␈α∞when␈↓ β|␈ε(p␈↓ ∧,␈ε"<␈ε(␈α
n␈ε6␈α
∀␈↓ ¬2␈ε(p␈↓ ¬]␈ε"+␈↓ ε
␈ε(q␈↓ ε≡␈ε");␈α∞and␈α∞the␈α∞pairs␈α∞of␈α∞such␈α∞occurrences␈α∞with
␈β
v␈↓ ↓]␈ε+N␈↓ ∧∂␈ε%2␈↓ ¬E␈ε%1
␈β∃␈↓ ↓H␈ε(N␈↓ ↓u␈ε"<␈↓ α#␈ε(p␈↓ αO␈ε"<␈↓ α⎇␈ε(p␈↓ β)␈ε"<␈↓ βW␈ε(N␈↓ ∧α␈ε"+␈↓ ∧.␈ε(q␈↓ ∧M␈ε"are␈αcoun␈α␈ted␈↓ ε⊂␈ε(N␈↓ ε;␈ε"+␈↓ εg␈ε(q␈↓ πα␈ε6␈␈↓ π.␈ε(p␈↓ π]␈ε"times.
␈β!␈↓ α6␈ε%1␈↓ β⊂␈ε%2␈↓ πA␈ε%2
␈βB␈↓ α␈ε"Let␈↓ αN␈ε(d␈↓ αl␈ε"(␈ε(n␈ε")␈αbe␈αthe␈αn␈α␈um␈α␈ber␈αof␈αpairs␈αof␈αoccurrences␈αof␈ε(␈αx␈ε"␈αin␈αpositions␈↓ 
ε␈ε(p␈↓ 
)␈ε",␈↓ 
9␈ε(p␈↓ 
f␈ε"with
␈βO␈↓ αa␈ε+t␈↓ 
→␈ε%1␈↓ 
L␈ε%2
␈βm␈↓ ↓H␈ε(p␈↓ ↓r␈ε"+␈ε(␈αλt␈ε"␈α
=␈↓ αc␈ε(p␈↓ β∂␈ε"<␈ε(␈α
n␈ε".␈α⊂The␈αanalysis␈αabo␈α␈v␈α␈e␈αsh␈α↓o␈α␈ws␈αthat
␈βz␈↓ ↓[␈ε%1␈↓ αv␈ε%2
␈β)␈↓ ↓u␈ε↓X␈↓ εn␈ε↓X
␈βK␈↓ αW␈ε"(␈↓ αc␈ε(q␈↓ α}␈ε6␈␈ε(␈αλmt␈ε")␈↓ βc␈ε(d␈↓ ∧≥␈ε"(␈↓ ∧)␈ε(N␈↓ ∧T␈ε"+␈↓ ¬␈ε(q␈↓ ¬∪␈ε")␈ε6␈α
∃␈↓ ¬W␈ε(S␈↓ ε∩␈ε6∃␈↓ πO␈ε"(␈↓ π[␈ε(q␈↓ πw␈ε6␈␈ε(␈αλm␈α␈t␈ε"␈α↓)␈↓ λ\␈ε(d␈↓ 	∃␈ε"(␈↓ 	!␈ε(N␈↓ 	D␈ε").␈↓ 
p␈ε"(20)
␈βW␈↓ βw␈ε+m␈α␈t␈↓ ¬l␈ε+N␈↓ λo␈ε+m␈α␈t
␈β{␈↓ ↓H␈ε%0<␈ε+␈α␈t␈ε%␈α↓<␈↓ α≥␈ε+q␈↓ α-␈ε%/␈ε+␈α↓m␈↓ ε@␈ε%0<␈ε+t␈ε%<␈↓ π⊗␈ε+q␈↓ π&␈ε%/␈ε+m
␈β
M␈↓ ↓H␈ε"Since␈αthe␈αoriginal␈αsequence␈αis␈↓ ¬≡␈ε(q␈↓ ¬1␈ε"-distributed,
␈β∞!␈↓ ¬Z␈ε"1
␈β∞1␈↓ πY␈ε%2␈ε+␈α↓m
␈β∞9␈↓ ¬β␈ε"lim␈↓ ¬y␈ε(d␈↓ ε2␈ε"(␈↓ ε>␈ε(N␈↓ εa␈ε")␈α
=␈α
1/␈↓ πI␈ε(b␈↓ 
p␈ε"(21)
␈β∞E␈↓ ε␈ε+mt
␈β∞M␈↓ ¬R␈∧∞M¬Rα#
␈β∞R␈↓ ¬R␈ε(N
␈β∞Y␈↓ ∧q␈ε+N␈↓ ¬
␈ε9!␈α␈1
␈β∂'␈↓ ↓H␈ε"for␈αall␈ε(␈αt␈ε",␈α0␈α
<␈ε(␈α
t␈ε"␈α
<␈↓ βd␈ε(q␈↓ βw␈ε"/␈ε(m␈ε",␈αand␈αtherefore␈αby␈α(20)␈αw␈α␈e␈αha␈α␈v␈α␈e
␈β∂t␈↓ ∧r␈ε↓X
␈β∂}␈↓ βW␈ε(S
␈β⊂
␈↓ βm␈ε+N
␈β⊂∞␈↓ πα␈ε%2␈ε+m␈↓ 	J␈ε%2␈ε+m
␈β⊂⊗␈↓ β	␈ε"lim␈↓ ∧⊗␈ε"=␈↓ ¬T␈ε"(␈↓ ¬`␈ε(q␈↓ ¬{␈ε6␈␈ε(␈αλmt␈ε")/␈↓ εr␈ε(b␈↓ π6␈ε"=␈↓ πd␈ε(q␈↓ πw␈ε"(␈↓ λβ␈ε(q␈↓ λ≡␈ε6␈␈ε(␈αλm␈ε")/2␈ε(m␈↓ 	:␈ε(b␈↓ 	t␈ε".␈↓ 
p␈ε"(22)
␈β⊂*␈↓ βW␈∧⊂*βWα2
␈β⊂0␈↓ β↑␈ε(N
␈β⊂6␈↓ αv␈ε+N␈↓ β∩␈ε9!1
␈β⊂F␈↓ ∧D␈ε%0<␈ε+t␈ε%<␈↓ ¬~␈ε+q␈↓ ¬*␈ε%/␈ε+m
␈β⊃_␈↓ ↓H␈ε"This␈αfact␈αwill␈αpro␈α␈v␈α␈e␈αthe␈αtheorem,␈αa$er␈αsome␈αmanipulation.
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"151
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα$␈↓ α␈ε"By␈αde|nition,
␈βαF␈↓ β3␈ε↓X␈↓ ∧S␈ε↓X
␈βαI␈↓ ¬'␈ε↓␈␈↓ 
]␈ε↓↓
␈βα`␈↓ πg␈ε%2
␈βαh␈↓ ↓␈␈ε"2␈↓ α⊃␈ε(S␈↓ αL␈ε"=␈↓ ¬5␈ε"(␈↓ ¬A␈ε(↔␈↓ ¬c␈ε"(␈ε(n␈ε"␈α␈)␈ε6␈αλ␈␈↓ εD␈ε(↔␈↓ εf␈ε"(␈ε(n␈ε6␈αλ␈␈↓ π<␈ε(q␈↓ πO␈ε")␈↓ π[␈ε")␈↓ π}␈ε6␈␈ε"␈αλ(␈↓ λ6␈ε(↔␈↓ λX␈ε"(␈ε(n␈ε")␈ε6␈αλ␈␈↓ 	:␈ε(↔␈↓ 	\␈ε"(␈ε(n␈ε6␈αλ␈␈↓ 
2␈ε(q␈↓ 
E␈ε"))␈↓ 
k␈ε",
␈βαt␈↓ α'␈ε+N␈↓ ¬S␈ε+j␈↓ εW␈ε+j␈↓ λI␈ε+j␈↓ 	M␈ε+j
␈ββ_␈↓ αz␈ε%1␈ε9∀␈ε+n␈ε9∀␈↓ βW␈ε+N␈↓ βs␈ε%+␈↓ ∧⊃␈ε+q␈↓ ∧3␈ε%0␈ε9∀␈↓ ∧←␈ε+j␈↓ ∧o␈ε%<␈ε+␈α␈m
␈ββQ␈↓ ↓H␈ε"and␈αw␈α␈e␈αcan␈αrem␈α↓o␈α␈v␈α␈e␈αthe␈αunsquared␈αterms␈αby␈αapplying␈α(16)␈αto␈αget
␈β∧∂␈↓ ∧n␈ε(T
␈β∧≠␈↓ ¬β␈ε+N
␈β∧∨␈↓ π/␈ε%2␈ε+m␈↓ λB␈ε+m
␈β∧'␈↓ ∧ ␈ε"lim␈↓ ¬-␈ε"=␈↓ ¬[␈ε(q␈↓ ¬n␈ε"(␈↓ ¬z␈ε(q␈↓ ε∃␈ε6␈␈ε(␈αλm␈ε")/␈ε(m␈↓ π∨␈ε(b␈↓ πa␈ε"+␈↓ λ
␈ε(q␈↓ λ ␈ε"/␈↓ λ2␈ε(b␈↓ λ]␈ε",␈↓ 
p␈ε"(23)
␈β∧;␈↓ ∧n␈∧∧;∧nα1
␈β∧A␈↓ ∧u␈ε(N
␈β∧G␈↓ ∧∞␈ε+N␈↓ ∧)␈ε9!1
␈β∧y␈↓ ↓H␈ε"where
␈β¬α␈↓ ∧i␈ε↓X␈↓ ε	␈ε↓X
␈β¬ε␈↓ ε]␈ε↓␈␈↓ 	¬␈ε↓↓
␈β¬⊗␈↓ 	∪␈ε%2
␈β¬$␈↓ βH␈ε(T␈↓ ∧α␈ε"=␈↓ εk␈ε(↔␈↓ π
␈ε"(␈ε(n␈ε"␈α␈)␈ε6␈αλ␈␈↓ πn␈ε(↔␈↓ λ⊂␈ε"(␈ε(n␈ε6␈αλ␈␈↓ λf␈ε(q␈↓ λy␈ε")␈↓ 	"␈ε".
␈β¬1␈↓ β\␈ε+N␈↓ ε⎇␈ε+j␈↓ λ↓␈ε+j
␈β¬U␈↓ ∧0␈ε%1␈ε9∀␈ε+n␈ε9∀␈↓ ¬
␈ε+N␈↓ ¬)␈ε%+␈↓ ¬F␈ε+q␈↓ ¬h␈ε%0␈ε9␈α↓∀␈↓ ε∃␈ε+j␈↓ ε$␈ε%<␈ε+m
␈βε∧␈↓ α␈ε"Using␈αthe␈αinequalit␈α␈y
␈βε>␈↓ ε9␈ε%2
␈βε@␈↓ ¬.␈ε↓X␈↓ π~␈ε↓X
␈βεI␈↓ ∧}␈ε↓∩␈↓ ε#␈ε↓∪
␈βεK␈↓ ∧h␈ε"1
␈βε[␈↓ λ↓␈ε%2
␈βεc␈↓ ε↓␈ε(a␈↓ εS␈ε6∀␈↓ πn␈ε(a
␈βεo␈↓ ε∀␈ε+j
␈βεs␈↓ λ↓␈ε+j
␈βεw␈↓ ∧h␈∧εw∧hα∩
␈βε|␈↓ ∧i␈ε(r
␈βπ∪␈↓ ¬∀␈ε%1␈ε9∀␈↓ ¬A␈ε+j␈↓ ¬P␈ε9∀␈↓ ¬m␈ε+r␈↓ π↓␈ε%1␈ε9∀␈↓ π-␈ε+j␈↓ π=␈ε9∀␈↓ πZ␈ε+r
␈βπL␈↓ ↓H␈ε"(cf.␈αexercise␈α1.2.3↑30),␈αw␈α␈e␈α|nd␈αthat
␈βλλ␈↓ 	1␈ε%2
␈βλ␈↓ β!␈ε↓X␈↓ ¬w␈ε↓X
␈βλ∞␈↓ εd␈ε↓␈␈↓ 	
␈ε↓↓
␈βλ∀␈↓ ¬(␈ε↓∩␈↓ 	≠␈ε↓∪
␈βλ∃␈↓ ∧I␈ε"1
␈βλ-␈↓ α
␈ε"lim␈αεsup␈↓ εr␈ε(↔␈↓ π∀␈ε"(␈ε(n␈ε")␈ε6␈αλ␈␈↓ πv␈ε(↔␈↓ λ_␈ε"(␈ε(n␈ε6␈αλ␈␈↓ λn␈ε(q␈↓ 	↓␈ε")
␈βλ9␈↓ π¬␈ε+j␈↓ λ	␈ε+j
␈βλA␈↓ β␈␈∧λAβ␈α↓&
␈βλF␈↓ β␈␈ε(N␈↓ ∧"␈ε"(␈↓ ∧.␈ε(N␈↓ ∧Y␈ε"+␈↓ ¬¬␈ε(q␈↓ ¬_␈ε")
␈βλT␈↓ α_␈ε+N␈↓ α4␈ε9!1
␈βλ]␈↓ β↓␈ε%0␈ε9∀␈↓ β.␈ε+j␈↓ β=␈ε%<␈ε+␈α␈m␈↓ ¬>␈ε%1␈ε9∀␈ε+n␈ε9∀␈↓ ε≠␈ε+N␈↓ ε7␈ε%+␈↓ εT␈ε+q
␈β	↓␈↓ λy␈ε%2␈ε+␈α↓m␈↓ 

␈ε+m
␈β		␈↓ εw␈ε6∀␈↓ π%␈ε(q␈↓ π9␈ε"(␈↓ πE␈ε(q␈↓ π`␈ε6␈␈ε(␈αλm␈ε"␈α␈)/␈ε(m␈↓ λi␈ε(b␈↓ 	+␈ε"+␈↓ 	W␈ε(q␈↓ 	j␈ε"/␈↓ 	|␈ε(b␈↓ 
'␈ε".␈↓ 
p␈ε"(24)
␈β	L␈↓ ↓H␈ε"We␈αalso␈αha␈α␈v␈α␈e
␈β	n␈↓ βB␈ε↓X␈↓ ¬|␈ε↓X
␈β	r␈↓ εj␈ε↓␈␈↓ 	∩␈ε↓↓
␈β
⊂␈↓ ↓U␈ε(q␈↓ ↓n␈ε(↔␈↓ α⊂␈ε"(␈↓ α≤␈ε(N␈↓ α?␈ε")␈ε6␈α
∀␈↓ ∧<␈ε(↔␈↓ ∧↑␈ε"(␈ε(n␈ε"␈α␈)␈α
=␈↓ εx␈ε(↔␈↓ π→␈ε"(␈ε(n␈ε")␈ε6␈αλ␈␈↓ π{␈ε(↔␈↓ λ≥␈ε"(␈ε(n␈ε6␈αλ␈␈↓ λs␈ε(q␈↓ 	ε␈ε")␈↓ 	*␈ε6∀␈↓ 	X␈ε(q␈↓ 	q␈ε(↔␈↓ 
∪␈ε"(␈↓ 
∨␈ε(N␈↓ 
J␈ε"+␈↓ 
v␈ε(q␈↓ 	␈ε"),
␈β
≥␈↓ α↓␈ε+j␈↓ ∧N␈ε+j␈↓ π
␈ε+j␈↓ λ∞␈ε+j␈↓ 
∧␈ε+j
␈β
A␈↓ ββ␈ε+N␈↓ β∨␈ε%<␈ε+␈α␈n␈ε9∀␈↓ βl␈ε+N␈↓ ∧λ␈ε%+␈↓ ∧&␈ε+q␈↓ ¬C␈ε%1␈ε9␈α↓∀␈ε+␈α␈n␈ε9∀␈↓ ε ␈ε+N␈↓ ε<␈ε%+␈↓ εZ␈ε+q
␈β
z␈↓ ↓H␈ε"and␈αputting␈αthis␈αin␈α␈to␈α(24)␈αgiv␈α␈es
␈β#␈↓ ∧⊃␈ε↓X
␈β&␈↓ ∧e␈ε↓␈␈↓ ε¬␈ε↓↓
␈β6␈↓ ε∪␈ε%2
␈β=␈↓ λ.␈ε%2␈ε+m␈↓ 	S␈ε+m
␈βE␈↓ α⎇␈ε"lim␈αεsup␈↓ ∧s␈ε(↔␈↓ ¬∃␈ε"(␈↓ ¬!␈ε(N␈↓ ¬D␈ε")/␈↓ ¬b␈ε(N␈↓ ε,␈ε6∀␈ε"␈α
(␈↓ εf␈ε(q␈↓ π↓␈ε6␈␈ε(␈αλm␈ε")/␈↓ πk␈ε(q␈↓ π}␈ε(m␈↓ λ≡␈ε(b␈↓ λ`␈ε"+␈αλ1/␈↓ 	0␈ε(q␈↓ 	C␈ε(b␈↓ 	n␈ε".␈↓ 
p␈ε"(25)
␈βQ␈↓ ¬¬␈ε+j
␈βl␈↓ βλ␈ε+N␈↓ β$␈ε9!␈α␈1
␈βu␈↓ βq␈ε%0␈ε9∀␈↓ ∧≥␈ε+j␈↓ ∧-␈ε%<␈ε+␈α␈m
␈β,␈↓ ↓H␈ε"This␈α
form␈α␈ula␈α
has␈α
been␈α
established␈α
whenev␈α␈er␈↓ π∂␈ε(q␈↓ π/␈ε"is␈α∞a␈α
m␈α␈ultiple␈α
of␈ε(␈α
m␈ε";␈α
and␈α
if␈α
w␈α␈e␈α
let
␈βW␈↓ ↓H␈ε(q␈↓ ↓e␈ε6!␈α
1␈ε"␈αw␈α␈e␈αobtain␈α(17),␈αcompleting␈αthe␈αpro␈α↓of.
␈β
α␈↓ α␈ε"F␈α⎇or␈αa␈αpossibly␈αsimpler␈αpro␈α↓of,␈αsee␈αJ.␈αW.␈αS.␈↓ π≠␈ε"Cassels,␈ε/␈αPaci|c␈αJ.␈αMath.␈ε2␈α2␈ε"␈α(1952),
␈β
.␈↓ ↓H␈ε"555↑557.
␈β
7␈↓ αw␈∧
7αw≠∂
␈β
f␈↓ α␈ε"Exercises␈α⊂29␈α∂and␈α⊂30␈α∂illustrate␈α⊂the␈α⊂n␈α↓on␈α␈trivialit␈α␈y␈α∂of␈α⊂this␈α∂theorem,␈α⊃and␈α∂they
␈β∞⊃␈↓ ↓H␈ε"also␈α∞dem␈α↓onstrate␈α∞the␈α∂fact␈α∞that␈α∞a␈↓ ¬←␈ε(q␈↓ ¬r␈ε"-distributed␈α∞sequence␈α∂will␈α∞ha␈α␈v␈α␈e␈α∞probabilities
␈β∞=␈↓ ↓H␈ε"deviating␈αfrom␈αthe␈αtrue␈α
(␈ε(m␈ε"␈α␈,␈ε(␈αεm␈ε")-distribution␈αprobabilities␈α
by␈αessen␈α␈tially␈α1/␈↓ 
m␈ε(q␈↓ ␈ε"at
␈β∞?␈↓ 
O␈ε6p
␈β∞B␈↓ 
m␈∧∞B
mα∀
␈β∞I␈↓ α0␈ε↓␈␈↓ β>␈ε↓↓
␈β∞h␈↓ ↓H␈ε"m␈α↓ost.␈↓ α>␈ε"Cf.␈α
(25).␈↓ β]␈ε"The␈α	full␈α
h␈α␈ypothesis␈α
of␈ε6␈α
1␈ε"-distribution␈α	is␈α
necessary␈α
for␈α
the␈α	pro␈α↓of
␈β∂∪␈↓ ↓H␈ε"of␈αthe␈αtheorem.
␈β∂>␈↓ α␈ε"As␈α⊂a␈α⊂result␈α⊂of␈α⊂Theorem␈α⊂C␈↓ ¬2␈ε",␈α⊃w␈α␈e␈α⊂can␈α⊂pro␈α␈v␈α␈e␈α⊂that␈α⊂an␈ε6␈α∂1␈ε"-distributed␈α⊂sequence
␈β∂i␈↓ ↓H␈ε"passes␈αthe␈↓ αq␈ε"serial␈αtest,␈αthe␈↓ ∧\␈ε"maxim␈α␈um-of-␈ε(t␈ε"␈αtest,␈αthe␈↓ πV␈ε"collision␈αtest,␈αand␈αthe␈αtests␈αon
␈β⊂∃␈↓ ↓H␈ε"subsequences␈α	men␈α␈tioned␈α	in␈α	Section␈α	3.3.2.␈α∂It␈α	is␈α	n␈α↓ot␈α	hard␈α	to␈α	sh␈α↓o␈α␈w␈α	that␈α	the␈↓ 
'␈ε"gap␈α	test,
␈β⊂@␈↓ ↓H␈ε"the␈↓ απ␈ε"pok␈α␈er␈α
test,␈α
and␈α
the␈↓ ∧B␈ε"run␈α
test␈α
are␈α
also␈α
satis|ed␈α
(see␈α
exercises␈α
12␈αthrough␈α
14);
␈β⊂k␈↓ ↓H␈ε"the␈↓ α¬␈ε"coupon␈αcollector's␈αtest␈αis␈αconsiderably␈αm␈α↓ore␈αdi}cult␈αto␈αdeal␈αwith,␈αbut␈αit␈αto␈α↓o
␈β⊃⊗␈↓ ↓H␈ε"is␈αsatis|ed␈α(see␈αexercises␈α15␈αand␈α16).
␈β∪(

␈β↓U␈↓ ↓H␈ε"152␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα$␈↓ α␈ε"The␈αexistence␈αof␈ε6␈α1␈ε"-distributed␈αsequences␈αof␈αa␈αrather␈αsimple␈αt␈α␈ype␈αis␈αguaran-
␈βαO␈↓ ↓H␈ε"teed␈αby␈αthe␈αnext␈αtheorem.
␈ββ⊃␈↓ ↓H␈ε2Theorem␈αF␈ε"␈α(J.␈↓ β8␈ε"F␈α⎇ranklin)␈ε2.␈ε/␈α→The␈↓ ¬6␈ε"[␈↓ ¬F␈ε"0,␈αε1)␈ε/␈αsequence␈↓ π&␈ε(U␈↓ πN␈ε/,␈↓ πd␈ε(U␈↓ λ␈ε/,␈↓ λ"␈ε".␈αε.␈αε.␈↓ λR␈ε/,␈αwith
␈ββ≥␈↓ π?␈ε%0␈↓ π⎇␈ε%1
␈ββ`␈↓ ε@␈ε+n
␈ββh␈↓ ¬?␈ε(U␈↓ ¬{␈ε"=␈↓ ε/␈ε(∩␈↓ εY␈ε"mod␈↓ π#␈ε"1␈↓ 
p␈ε"(26)
␈ββt␈↓ ¬X␈ε+n
␈β∧?␈↓ ↓H␈ε/is␈ε6␈α1␈ε/-distributed␈αfor␈αalm␈α↓ost␈αall␈αreal␈αn␈α␈um␈α␈bers␈↓ π	␈ε(∩␈↓ π$␈ε">␈α
1␈ε/.␈α⊂That␈αis,␈αthe␈αset
␈β¬∃␈↓ βo␈ε6f␈↓ ∧π␈ε(∩␈↓ ∧#␈ε6j␈↓ ∧7␈ε(∩␈↓ ∧R␈ε">␈α
1␈↓ ¬≡␈ε"and␈α(26)␈αis␈αn␈α↓ot␈ε6␈α1␈ε"-distributed␈↓ λs␈ε6g
␈β¬l␈↓ ↓H␈ε/is␈αof␈αmeasure␈αzero.
␈βε.␈↓ ↓H␈ε"The␈αλpro␈α↓ofs␈α	of␈α	this␈α	theorem␈αλand␈α	some␈α	generalizations␈α	are␈αλgiv␈α␈en␈α	in␈α	F␈α⎇ranklin's␈αλpaper
␈βεY␈↓ ↓H␈ε"cited␈αbelo␈α␈w.
␈βεb␈↓ β*␈∧εbβ*≠∂
␈βπ≠␈↓ α␈ε"F␈α⎇ranklin␈α
has␈α
sh␈α↓o␈α␈wn␈α
that␈↓ ¬≥␈ε(∩␈↓ ¬;␈ε"m␈α␈ust␈α
be␈α
a␈α
transcenden␈α␈tal␈α
n␈α␈um␈α␈ber␈α
for␈α
(26)␈α
to␈αbe
␈βπ@␈↓ ¬ ␈ε+n
␈βπF␈↓ ↓H␈ε61␈ε"-distributed.␈α⊂The␈αpo␈α␈w␈α␈ers␈α(␈↓ ¬
␈ε(→␈↓ ¬8␈ε"mod␈↓ εα␈ε"1)␈αha␈α␈v␈α␈e␈αbeen␈αlaboriously␈α
computed␈αfor␈ε(␈αn␈ε6␈α	∀
␈βπq␈↓ ↓H␈ε"10000,␈↓ α;␈ε"using␈α∂m␈α␈ultiple-precision␈α∂arithmetic,␈α⊂and␈α∂the␈α∂m␈α↓ost␈α∂signi|can␈α␈t␈α∂35␈α∂bits␈α∂of
␈βλ≤␈↓ ↓H␈ε"each␈α∂of␈α∂these␈α∂n␈α␈um␈α␈bers,␈α⊂stored␈α⊂on␈α∂a␈α∂disk␈α∂|le,␈α⊂ha␈α␈v␈α␈e␈α⊂successfully␈α∂been␈α∂used␈α∂as␈α∂a
␈βλH␈↓ ↓H␈ε"source␈α∞of␈α∞uniform␈α
deviates.␈α⊗According␈α∞to␈α∞Theorem␈α∞F␈↓ λ_␈ε",␈α∂the␈α∞probabilit␈α␈y␈α
that␈α∞the
␈βλm␈↓ α↑␈ε+n
␈βλs␈↓ ↓H␈ε"po␈α␈w␈α␈ers␈α
(␈↓ αH␈ε(→␈↓ αw␈ε"mod␈↓ βA␈ε"1)␈α
are␈ε6␈α1␈ε"-distributed␈αis␈α
equal␈αto␈α1;␈αy␈α␈et␈α
because␈αthere␈αare␈α
uncoun-
␈β	≡␈↓ ↓H␈ε"tably␈α	man␈α␈y␈α	real␈α	n␈α␈um␈α␈bers,␈α
this␈α	giv␈α␈es␈α	us␈α	n␈α↓o␈α	information␈α	as␈α	to␈α	whether␈α	the␈α	sequence
␈β	I␈↓ ↓H␈ε"is␈αreally␈ε6␈α1␈ε"-distributed␈αor␈αn␈α↓ot.␈α∂It␈αis␈αa␈αfairly␈αsafe␈αbet␈αthat␈αn␈α↓obody␈αin␈αour␈αlifetimes
␈β	t␈↓ ↓H␈ε"will␈α
ev␈α␈er␈ε/␈α
pro␈α␈v␈α␈e␈ε"␈α
that␈α
this␈α
particular␈α
sequence␈α
is␈ε/␈α
n␈α↓ot␈ε6␈α∞1␈ε"-distributed;␈α
but␈α
it␈α
migh␈α␈t
␈β
 ␈↓ ↓H␈ε"n␈α↓ot␈αbe.␈α⊂Because␈αof␈αthese␈αconsiderations,␈αone␈αma␈α␈y␈αlegitimately␈αw␈α␈onder␈αif␈αthere␈αis
␈β
K␈↓ ↓H␈ε"an␈α␈y␈ε/␈α	explicit␈ε"␈α	sequence␈αλthat␈α	is␈ε6␈α	1␈ε"-distributed;␈α
i.e.,␈ε/␈α
is␈α	there␈α	an␈α	algorithm␈α	to␈αλcompute
␈β
v␈↓ ↓H␈ε/real␈α∞n␈α␈um␈α␈bers␈↓ β#␈ε(U␈↓ β]␈ε/for␈α∞all␈ε(␈α∂n␈ε6␈α∞∃␈ε"␈α∞0␈ε/,␈α⊂such␈α∞that␈α∂the␈α∞sequence␈ε6␈α∂h␈↓ λV␈ε(U␈↓ 	↓␈ε6i␈ε/␈α∂is␈ε6␈α∞1␈ε/-distributed?
␈βα␈↓ β<␈ε+n␈↓ λo␈ε+n
␈β!␈↓ ↓H␈ε"The␈αansw␈α␈er␈αis␈α
y␈α␈es,␈αas␈αsh␈α↓o␈α␈wn␈α
for␈αexample␈α
by␈αD.␈αE.␈↓ πj␈ε"Kn␈α␈uth␈αin␈ε/␈αBIT␈ε2␈α
5␈ε"␈α(1965),␈α246↑
␈βL␈↓ ↓H␈ε"250.␈α~The␈α∂sequence␈α⊂constructed␈α∂there␈α⊂consists␈α∂en␈α␈tirely␈α∂of␈α⊂rational␈α∂n␈α␈um␈α␈bers;␈α⊃in
␈βx␈↓ ↓H␈ε"fact,␈α⊂each␈α⊂n␈α␈um␈α␈ber␈↓ βz␈ε(U␈↓ ∧5␈ε"has␈α⊂a␈α⊂terminating␈α⊂represen␈α␈tation␈α⊂in␈α⊂the␈α⊂binary␈α∂n␈α␈um␈α␈ber
␈β∧␈↓ ∧∪␈ε+n
␈β#␈↓ ↓H␈ε"system.␈α∩An␈α↓other␈αconstruction␈α
of␈α
an␈αexplicit␈ε6␈α
1␈ε"-distributed␈α
sequence,␈αsomewhat
␈βN␈↓ ↓H␈ε"m␈α↓ore␈αcomplicated␈αthan␈αthe␈αsequence␈αjust␈αcited,␈αfollo␈α␈ws␈αfrom␈αTheorem␈αW␈αbelo␈α␈w.
␈βy␈↓ ↓H␈ε"See␈αalso␈αN.␈αM.␈↓ β8␈ε"Korobo␈α␈v,␈ε/␈αIzv.␈αAk␈α⎇ad.␈αNauk␈αSSSR␈ε2␈α20␈ε"␈α(1956),␈α649↑660.
␈β
;␈↓ ↓H␈ε2C.␈α∪Does␈ε6␈α∪1␈ε2-distributed␈ε"␈α∪=␈ε2␈α∀random?␈ε"␈α&In␈α∪view␈α∪of␈α∀all␈α∪the␈α∪abo␈α␈v␈α␈e␈α∪theory␈α∪about
␈β
f␈↓ ↓H␈ε61␈ε"-distributed␈α⊂sequences,␈α⊃w␈α␈e␈α⊂can␈α⊂be␈α⊃sure␈α⊂of␈α⊂one␈α⊂thing:␈α∃the␈α⊂concept␈α⊂of␈α⊂an␈ε6␈α⊂1␈ε"-
␈β∞⊃␈↓ ↓H␈ε"distributed␈α
sequence␈α
is␈α
an␈α
importan␈α␈t␈α
one␈α
in␈α
mathematics.␈α∪There␈α
is␈α
also␈α
a␈α
go␈α↓od
␈β∞<␈↓ ↓H␈ε"deal␈αof␈αevidence␈αthat␈αthe␈αfollo␈α␈wing␈αstatemen␈α␈t␈αis␈αa␈αv␈α⎇alid␈αform␈α␈ulation␈αof␈αthe␈αin␈α␈tui-
␈β∞h␈↓ ↓H␈ε"tiv␈α␈e␈αidea␈αof␈αrandomness:
␈β∂)␈↓ ↓H␈ε2De|nition␈α∩R1.␈ε/␈α&A␈↓ ∧↓␈ε"[␈↓ ∧⊃␈ε"0,␈αε1)␈ε/␈α∪sequence␈α∪is␈α∪de|ned␈α∪to␈α∩be␈α∪\random"␈α∪if␈α∪it␈α∪is␈α∪an␈ε6␈α∩1␈ε/-
␈β∂U␈↓ ↓H␈ε/distributed␈αsequence.
␈β⊂⊗␈↓ ↓H␈ε"We␈α
ha␈α␈v␈α␈e␈α
seen␈α
that␈α
sequences␈α
meeting␈α
this␈α
de|nition␈α
will␈α
satisfy␈α
all␈α
the␈α
statistical
␈β⊂B␈↓ ↓H␈ε"tests␈αof␈αSection␈α3.3.2␈αand␈αman␈α␈y␈αm␈α↓ore.
␈β⊂m␈↓ α␈ε"Let␈α∞us␈α
attempt␈α∞to␈α
criticize␈α∞this␈α
de|nition␈α∞objectiv␈α␈ely.␈α∃First␈α
of␈α∞all,␈α∞is␈α
ev␈α␈ery
␈β⊃_␈↓ ↓H␈ε"\truly␈α	random"␈α	sequence␈ε6␈α	1␈ε"-distributed?␈α∂There␈α
are␈α	uncoun␈α␈tably␈α	man␈α␈y␈α	sequences
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"153
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα$␈↓ ↓H␈ε(U␈↓ ↓p␈ε",␈↓ α	␈ε(U␈↓ α1␈ε",␈↓ αK␈ε".␈αε.␈αε.␈↓ β∧␈ε"of␈α∂real␈α∂n␈α␈um␈α␈bers␈α∂bet␈α␈w␈α␈een␈α∂zero␈α∂and␈α∂one.␈α→If␈α∂a␈α∂truly␈α∂random␈α∂n␈α␈um␈α␈ber
␈βα0␈↓ ↓`␈ε%0␈↓ α"␈ε%1
␈βαO␈↓ ↓H␈ε"generator␈α∂is␈α∂sampled␈α∂to␈α∂giv␈α␈e␈α∂v␈α⎇alues␈↓ ε
␈ε(U␈↓ ε2␈ε",␈↓ εL␈ε(U␈↓ εt␈ε",␈↓ π∞␈ε".␈αε.␈αε.␈↓ π>␈ε",␈α⊂an␈α␈y␈α∂of␈α∂the␈α∂possible␈α∂sequences
␈βα\␈↓ ε#␈ε%0␈↓ εe␈ε%1
␈βαz␈↓ ↓H␈ε"ma␈α␈y␈α
be␈α∞considered␈α∞equally␈α∞lik␈α␈ely,␈α∞and␈α∞some␈α∞of␈α∞the␈α
sequences␈α∞(indeed,␈α∞uncoun␈α␈t-
␈ββ&␈↓ ↓H␈ε"ably␈αman␈α␈y␈αof␈αthem)␈αare␈αn␈α↓ot␈αev␈α␈en␈αequidistributed.␈α∂On␈αthe␈αother␈αhand,␈αusing␈αan␈α␈y
␈ββQ␈↓ ↓H␈ε"reasonable␈αde|nition␈α
of␈α
probabilit␈α␈y␈α
on␈α
this␈αspace␈α
of␈α
all␈α
possible␈α
sequences␈αleads
␈ββ|␈↓ ↓H␈ε"us␈α∞to␈α∂conclude␈α∞that␈α∂a␈α∂random␈α∞sequence␈α∂is␈ε6␈α∂1␈ε"-distributed␈ε/␈α∞with␈α∂probabilit␈α␈y␈α∞one.
␈β∧'␈↓ ↓H␈ε"We␈α
are␈α
therefore␈α
led␈α∞to␈α
formalize␈α
F␈α⎇ranklin's␈α
de|nition␈α∞of␈α
randomness␈α
(as␈α
giv␈α␈en
␈β∧R␈↓ ↓H␈ε"at␈αthe␈αbeginning␈αof␈αthis␈αsection)␈αin␈αthe␈αfollo␈α␈wing␈αw␈α␈a␈α␈y:
␈β¬∞␈↓ ↓H␈ε2De|nition␈αR2.␈ε/␈α→A␈↓ βf␈ε"[␈↓ βv␈ε"0,␈αε1)␈ε/␈αsequence␈ε6␈αh␈↓ ¬a␈ε(U␈↓ ε
␈ε6i␈ε/␈αis␈αde|ned␈αto␈αbe␈α\random"␈αif,␈αwhenev␈α␈er␈↓ ⊃␈ε(P
␈β¬~␈↓ ¬z␈ε+n
␈β¬9␈↓ ↓H␈ε/is␈α∞a␈α∞propert␈α␈y␈α∞such␈α∞that␈↓ ∧F␈ε(P␈↓ ∧b␈ε"(␈ε6h␈↓ ∧z␈ε(V␈↓ ¬"␈ε6i␈ε")␈ε/␈α∞h␈α↓olds␈α∞with␈α∞probabilit␈α␈y␈α∂one␈α∞for␈α∞a␈α∞sequence␈ε6␈α∞h␈↓ 
x␈ε(V␈↓  ␈ε6i
␈β¬E␈↓ ¬⊂␈ε+n␈↓ ∞␈ε+n
␈β¬d␈↓ ↓H␈ε/of␈αindependen␈α␈t␈αsamples␈αof␈αrandom␈αv␈α⎇ariables␈αfrom␈αthe␈αuniform␈αdistribution,␈αthen
␈βε∂␈↓ ↓H␈ε(P␈↓ ↓c␈ε"(␈ε6h␈↓ ↓{␈ε(U␈↓ α&␈ε6i␈ε")␈ε/␈α
is␈αtrue.
␈βε≤␈↓ α∀␈ε+n
␈βεK␈↓ α␈ε"Is␈αit␈αperhaps␈αpossible␈αthat␈αDe|nition␈αR1␈αis␈αequiv␈α⎇alen␈α␈t␈αto␈αDe|nition␈αR2?␈α∂Let
␈βεv␈↓ ↓H␈ε"us␈α⊃try␈α⊃out␈α⊃some␈α⊃possible␈α∩objections␈α⊃to␈α⊃De|nition␈α⊃R1,␈α∪and␈α⊃see␈α⊃whether␈α⊃these
␈βπ!␈↓ ↓H␈ε"criticisms␈αare␈αv␈α⎇alid.
␈βπL␈↓ α␈ε"In␈α∂the␈α∞|rst␈α∂place,␈α∂De|nition␈α∞R1␈α∂deals␈α∂only␈α∞with␈α∂limiting␈α∞properties␈α∂of␈α∞the
␈βπw␈↓ ↓H␈ε"sequence␈α	as␈ε(␈α	n␈ε6␈α
!␈α
1␈ε".␈α∂There␈α
are␈ε6␈α	1␈ε"-distributed␈α	sequences␈α
in␈α	which␈α
the␈α	|rst␈α	million
␈βλ#␈↓ ↓H␈ε"elemen␈α␈ts␈αare␈αall␈αzero;␈αsh␈α↓ould␈αsuch␈αa␈αsequence␈αbe␈αconsidered␈αrandom?
␈βλN␈↓ α␈ε"This␈α⊂objection␈α⊂is␈α∂n␈α↓ot␈α⊂v␈α␈ery␈α⊂v␈α⎇alid.␈α≠If␈ε(␈α⊂∂␈ε"␈α∂is␈α⊂an␈α␈y␈α⊂positiv␈α␈e␈α∂n␈α␈um␈α␈ber,␈α⊃there␈α⊂is␈α∂n␈α↓o
␈βλy␈↓ ↓H␈ε"reason␈αwh␈α␈y␈αthe␈α|rst␈αmillion␈αelemen␈α␈ts␈αof␈αa␈αsequence␈αsh␈α↓ould␈αn␈α↓ot␈αall␈αbe␈αless␈αthan␈ε(␈α∂␈ε"␈α␈.
␈β	$␈↓ ↓H␈ε"With␈α⊂probabilit␈α␈y␈α⊂one,␈α∩a␈α⊂truly␈α⊂random␈α⊂sequence␈α⊃con␈α␈tains␈α⊂in|nitely␈α⊂man␈α␈y␈α⊂runs
␈β	O␈↓ ↓H␈ε"of␈α∞a␈α∂million␈α∞consecutiv␈α␈e␈α∂elemen␈α␈ts␈α∂less␈α∞than␈ε(␈α∂∂␈ε",␈α∂so␈α∞wh␈α␈y␈α∂can't␈α∂this␈α∞happen␈α∂at␈α∞the
␈β	{␈↓ ↓H␈ε"beginning␈αof␈αthe␈αsequence?
␈β
&␈↓ α␈ε"On␈α
the␈α∞other␈α
hand,␈α∞consider␈α∞De|nition␈α
R2␈α
and␈α∞let␈↓ λ=␈ε(P␈↓ λf␈ε"be␈α∞the␈α
propert␈α␈y␈α
that
␈β
Q␈↓ ↓H␈ε"all␈αelemen␈α␈ts␈α
of␈α
the␈αsequence␈α
are␈α
distinct;␈↓ εV␈ε(P␈↓ ε}␈ε"is␈α
true␈αwith␈α
probabilit␈α␈y␈α
one,␈α
so␈αan␈α␈y
␈β
|␈↓ ↓H␈ε"sequence␈αwith␈αa␈αmillion␈αzeros␈αis␈αn␈α↓ot␈αrandom␈αby␈ε/␈αthis␈ε"␈αcriterion.
␈β'␈↓ α␈ε"No␈α␈w␈αlet␈↓ β∩␈ε(P␈↓ β9␈ε"be␈αthe␈αpropert␈α␈y␈αthat␈ε/␈αn␈α↓o␈ε"␈αelemen␈α␈t␈αof␈αthe␈αsequence␈αis␈αequal␈αto␈αzero;
␈βS␈↓ ↓H␈ε"again,␈↓ α3␈ε(P␈↓ α\␈ε"is␈α
true␈α
with␈α
probabilit␈α␈y␈α
one,␈α∞so␈α
by␈α
De|nition␈α
R2␈α
an␈α␈y␈α
sequence␈α
with␈α
a
␈β}␈↓ ↓H␈ε"zero␈α
elemen␈α␈t␈α
is␈α	n␈α↓onrandom.␈α⊂More␈α
generally,␈α
h␈α↓o␈α␈w␈α␈ev␈α␈er,␈α
let␈↓ λC␈ε(x␈↓ λq␈ε"be␈α
an␈α␈y␈α	|xed␈α
n␈α␈um␈α␈ber
␈β
␈↓ λW␈ε%0
␈β)␈↓ ↓H␈ε"bet␈α␈w␈α␈een␈α
zero␈α
and␈α
one,␈αand␈α
let␈↓ ¬∨␈ε(P␈↓ ¬E␈ε"be␈α
the␈α
propert␈α␈y␈α
that␈αn␈α↓o␈α
elemen␈α␈t␈α
of␈α
the␈α
sequence
␈βT␈↓ ↓H␈ε"is␈α∂equal␈α∂to␈↓ α␈␈ε(x␈↓ β"␈ε";␈α⊃De|nition␈α∂R2␈α∂n␈α↓o␈α␈w␈α∂sa␈α␈ys␈α∂that␈α∂n␈α↓o␈α∂random␈α∂sequence␈α∂ma␈α␈y␈α∂con␈α␈tain
␈βa␈↓ β∪␈ε%0
␈β␈␈↓ ↓H␈ε"the␈α∞elemen␈α␈t␈↓ β∂␈ε(x␈↓ β2␈ε"!␈α↔We␈α∂can␈α∞n␈α↓o␈α␈w␈α∞pro␈α␈v␈α␈e␈α∂that␈ε/␈α∞n␈α↓o␈α∞sequence␈α∂satis|es␈α∞the␈α∞condition␈α∞of
␈β
␈↓ β#␈ε%0
␈β
+␈↓ ↓H␈ε/De|nition␈αR2.␈ε"␈α⊂(F␈α⎇or␈αif␈↓ ∧'␈ε(U␈↓ ∧O␈ε",␈↓ ∧e␈ε(U␈↓ ¬
␈ε",␈↓ ¬#␈ε".␈αε.␈αε.␈↓ ¬Y␈ε"is␈αsuch␈αa␈αsequence,␈αtak␈α␈e␈↓ λZ␈ε(x␈↓ 	π␈ε"=␈↓ 	5␈ε(U␈↓ 	]␈ε".)
␈β
7␈↓ ∧?␈ε%0␈↓ ∧⎇␈ε%1␈↓ λn␈ε%0␈↓ 	N␈ε%0
␈β
V␈↓ α␈ε"Therefore␈α∂if␈α∂R1␈α∂is␈α∂to␈α↓o␈α∂w␈α␈eak␈α∂a␈α∂de|nition,␈α⊂R2␈α∂is␈α∂certainly␈α∂to␈α↓o␈α∂strong.␈α→The
␈β∞↓␈↓ ↓H␈ε"\righ␈α␈t"␈α
de|nition␈α	m␈α␈ust␈α
be␈α
less␈α
strict␈α
than␈α
R2.␈α∂We␈α
ha␈α␈v␈α␈e␈α
n␈α↓ot␈α
really␈α
sh␈α↓o␈α␈wn␈α
that␈α	R1
␈β∞,␈↓ ↓H␈ε"is␈α
to␈α↓o␈α
w␈α␈eak,␈α
h␈α↓o␈α␈w␈α␈ev␈α␈er,␈α
so␈α
let␈α
us␈α
con␈α␈tin␈α␈ue␈α
to␈α
attack␈α
it␈α
some␈α
m␈α↓ore.␈α∩As␈α
men␈α␈tioned
␈β∞W␈↓ ↓H␈ε"abo␈α␈v␈α␈e,␈α∪an␈ε6␈α⊃1␈ε"-distributed␈α∩sequence␈α⊃of␈ε/␈α∩rational␈ε"␈α∩n␈α␈um␈α␈bers␈α⊃has␈α∩been␈α⊃constructed.
␈β∂β␈↓ ↓H␈ε"(Indeed,␈α
this␈αis␈α
n␈α↓ot␈α
so␈α
surprising;␈α
see␈α
exercise␈α
18.)␈α~Alm␈α↓ost␈α
all␈α
real␈α
n␈α␈um␈α␈bers␈αare
␈β∂.␈↓ ↓H␈ε"irrational;␈αperhaps␈αw␈α␈e␈αsh␈α↓ould␈αinsist␈αthat
␈β∂x␈↓ ¬␈ε"P␈↓ ¬#␈ε"r␈↓ ¬2␈ε"(␈↓ ¬>␈ε(U␈↓ ¬u␈ε"is␈αrational␈↓ π∪␈ε")␈α
=␈α
0
␈β⊂∧␈↓ ¬V␈ε+n
␈β⊂B␈↓ ↓H␈ε"for␈αa␈αrandom␈αsequence.
␈β⊂m␈↓ α␈ε"Note␈α∞that␈α∂the␈α∞de|nition␈α∞of␈α∞equidistribution␈α∂sa␈α␈ys␈α∞that␈↓ λj␈ε"P␈↓ 	α␈ε"r␈↓ 	⊃␈ε"(␈ε(u␈ε6␈α
∀␈↓ 	q␈ε(U␈↓ 
*␈ε"<␈↓ 
\␈ε(v␈↓ 
o␈ε")␈α
=
␈β⊂y␈↓ 

␈ε+n
␈β⊃_␈↓ ↓H␈ε(v␈↓ ↓f␈ε6␈␈ε(␈αu␈ε".␈α≡There␈α⊃is␈α⊂an␈α⊃obvious␈α⊃w␈α␈a␈α␈y␈α⊃to␈α⊂generalize␈α⊃this␈α⊃de|nition,␈α∩using␈α⊂measure
␈β∪(

␈β↓U␈↓ ↓H␈ε"154␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα"␈↓ ↓H␈ε"theory:␈α\If␈↓ αy␈ε(S␈↓ β≤␈ε6∩␈↓ βJ␈ε"[␈↓ βZ␈ε"0,␈αε1)␈αis␈αa␈αset␈αof␈αmeasure␈ε(␈α⊗␈ε",␈αthen
␈βαr␈↓ ¬7␈ε"P␈↓ ¬O␈ε"r␈↓ ¬]␈ε"(␈↓ ¬i␈ε(U␈↓ ε≡␈ε62␈↓ ε@␈ε(S␈↓ εZ␈ε")␈α
=␈ε(␈α
⊗␈ε"␈α␈,␈↓ 
p␈ε"(27)
␈βα␈␈↓ εα␈ε+n
␈ββC␈↓ ↓H␈ε"for␈α⊂all␈α⊂random␈α⊃sequences␈ε6␈α⊂h␈↓ ∧u␈ε(U␈↓ ¬!␈ε6i␈ε"."␈α≥In␈α⊂particular,␈α∩if␈↓ πr␈ε(S␈↓ λ≠␈ε"is␈α⊃the␈α⊂set␈α⊃of␈α⊂rationals,␈α⊃it
␈ββO␈↓ ¬∞␈ε+n
␈ββn␈↓ ↓H␈ε"has␈α
measure␈↓ β⊗␈ε"zero,␈α∞so␈α
n␈α↓o␈α
sequence␈α∞of␈α
rational␈α
n␈α␈um␈α␈bers␈α∞is␈α
equidistributed␈α
in␈α
this
␈β∧→␈↓ ↓H␈ε"generalized␈αsense.␈α⊂It␈αis␈αreasonable␈αto␈αexpect␈αthat␈αTheorem␈αB␈αcould␈αbe␈αextended
␈β∧D␈↓ ↓H␈ε"to␈↓ ↓r␈ε"Lebesgue␈α
in␈α␈tegration␈α
instead␈α
of␈α
Riemann␈α
in␈α␈tegration,␈α
if␈α
propert␈α␈y␈α
(27)␈α
is␈α
stipu-
␈β∧o␈↓ ↓H␈ε"lated.␈α Ho␈α␈w␈α␈ev␈α␈er,␈α∪once␈α⊃again␈α⊃w␈α␈e␈α∩|nd␈α⊃that␈α∩de|nition␈α⊃(27)␈α⊃is␈α∩to␈α↓o␈α⊃strict,␈α∪for␈ε/␈α⊃n␈α↓o
␈β¬≠␈↓ ↓H␈ε"sequence␈α⊂satis|es␈α⊂that␈α⊂propert␈α␈y.␈α≤If␈↓ ε∞␈ε(U␈↓ ε6␈ε",␈↓ εQ␈ε(U␈↓ εy␈ε",␈↓ π∀␈ε".␈αε.␈αε.␈↓ πN␈ε"is␈α⊂an␈α␈y␈α⊂sequence,␈α⊃the␈α⊂set␈↓ 
←␈ε(S␈↓ λ␈ε"=
␈β¬'␈↓ ε&␈ε%0␈↓ εi␈ε%1
␈β¬F␈↓ ↓H␈ε6f␈↓ ↓Z␈ε(U␈↓ αα␈ε",␈↓ α∩␈ε(U␈↓ α:␈ε",␈↓ αJ␈ε".␈αε.␈αε.␈↓ αz␈ε6g␈ε"␈α∂is␈α⊂of␈α⊂measure␈α∂zero,␈α⊃y␈α␈et␈↓ ε≠␈ε"P␈↓ ε3␈ε"r␈↓ εA␈ε"(␈↓ εM␈ε(U␈↓ πλ␈ε62␈↓ π0␈ε(S␈↓ πJ␈ε")␈α⊂=␈α⊂1.␈α≠Th␈α␈us,␈α⊂by␈α⊂the␈α⊂force␈α∂of
␈β¬R␈↓ ↓r␈ε%0␈↓ α*␈ε%1␈↓ εf␈ε+n
␈β¬q␈↓ ↓H␈ε"the␈α
same␈αargumen␈α␈t␈αw␈α␈e␈αused␈αto␈α
exclude␈αrationals␈αfrom␈αrandom␈αsequences,␈αw␈α␈e␈α
can
␈βε≤␈↓ ↓H␈ε"exclude␈αall␈αrandom␈αsequences.
␈βεG␈↓ α␈ε"So␈αfar␈αDe|nition␈αR1␈αhas␈αpro␈α␈v␈α␈ed␈αto␈αbe␈αdefensible.␈α⊂There␈αare,␈αh␈α↓o␈α␈w␈α␈ev␈α␈er,␈αsome
␈βεs␈↓ ↓H␈ε"quite␈α
v␈α⎇alid␈α∞objections␈α
to␈α∞it.␈α∀F␈α⎇or␈α
example,␈α∞if␈α∞w␈α␈e␈α
ha␈α␈v␈α␈e␈α∞a␈α
random␈α∞sequence␈α
in␈α
the
␈βπ≡␈↓ ↓H␈ε"in␈α␈tuitiv␈α␈e␈αsense,␈αthe␈αin|nite␈αsubsequence
␈βπn␈↓ ∧Y␈ε(U␈↓ ¬↓␈ε",␈↓ ¬→␈ε(U␈↓ ¬A␈ε",␈↓ ¬Y␈ε(U␈↓ ε↓␈ε",␈↓ ε→␈ε(U␈↓ εA␈ε",␈↓ εY␈ε".␈αε.␈αε.␈↓ π	␈ε",␈↓ π!␈ε(U␈↓ πY␈ε",␈↓ πq␈ε".␈αε.␈αε.␈↓ 
p␈ε"(28)
␈βπy␈↓ πL␈ε'2
␈βπz␈↓ ∧r␈ε%0␈↓ ¬2␈ε%1␈↓ ¬q␈ε%4␈↓ ε1␈ε%9
␈βπ{␈↓ π9␈ε+n
␈βλ>␈↓ ↓H␈ε"sh␈α↓ould␈αalso␈αbe␈αa␈αrandom␈αsequence.␈α⊂This␈αis␈αn␈α↓ot␈αalw␈α␈a␈α␈ys␈αtrue␈αfor␈αan␈ε6␈α1␈ε"-distributed
␈βλi␈↓ ↓H␈ε"sequence.␈α⊗In␈α∞fact,␈α∂if␈α∞w␈α␈e␈α∞tak␈α␈e␈α∂an␈α␈y␈ε6␈α∞1␈ε"-distributed␈α∞sequence␈α∞and␈α∞set␈↓ 	h␈ε(U␈↓ 
/␈ε6␈ ␈ε"␈α
0␈α∞for
␈βλu␈↓ 
∀␈ε'2
␈βλv␈↓ 
↓␈ε+n
␈β	∃␈↓ ↓H␈ε"all␈ε(␈αn␈ε",␈αthe␈αcoun␈α␈ts␈↓ βQ␈ε(↔␈↓ βu␈ε"(␈ε(n␈ε"␈α␈)␈αthat␈αappear␈αin␈αthe␈αtest␈αof␈↓ πE␈ε(k␈↓ πY␈ε"-distributivit␈α␈y␈αare␈αchanged␈αby
␈β	!␈↓ βd␈ε+k
␈β	?␈↓ αI␈ε6p
␈β	@␈↓ ↓H␈ε"at␈α
m␈α↓ost␈↓ αg␈ε(n␈↓ α|␈ε",␈αso␈α
the␈α
limits␈α
of␈α
the␈α	ratios␈↓ ε≡␈ε(↔␈↓ εB␈ε"(␈ε(n␈ε"␈α␈)/␈ε(n␈ε"␈α
remain␈α
unchanged.␈α∂De|nition␈α
R1
␈β	B␈↓ αg␈∧	Bαgα⊗
␈β	L␈↓ ε1␈ε+k
␈β	k␈↓ ↓H␈ε"unfortunately␈αfails␈αto␈αsatisfy␈αthis␈αrandomness␈αcriterion.
␈β
⊗␈↓ α␈ε"Perhaps␈αw␈α␈e␈αsh␈α↓ould␈αstrengthen␈αR1␈αas␈αfollo␈α␈ws:
␈β
T␈↓ ↓H␈ε2De|nition␈α∞R3.␈ε/␈α≡A␈↓ βp␈ε"[␈↓ ∧↓␈ε"0,␈αε1)␈ε/␈α∞sequence␈α∂is␈α∞said␈α∂to␈α∞be␈α∂\random"␈α∞if␈α∂each␈α∞of␈α∂its␈α∞in|nite
␈β
␈␈↓ ↓H␈ε/subsequences␈αis␈ε6␈α1␈ε/-distributed.
␈β=␈↓ ↓H␈ε"Once␈αλagain,␈α
h␈α↓o␈α␈w␈α␈ev␈α␈er,␈α	the␈α	de|nition␈αλturns␈α	out␈α	to␈α	be␈αλto␈α↓o␈α	strict;␈α
an␈α␈y␈αλequidistributed
␈βi␈↓ ↓H␈ε"sequence␈ε6␈αh␈↓ αh␈ε(U␈↓ β∪␈ε6i␈ε"␈αhas␈αa␈αm␈α↓on␈α↓otonic␈αsubsequence␈αwith␈↓ πW␈ε(U␈↓ λ∃␈ε"<␈↓ λC␈ε(U␈↓ 	↓␈ε"<␈↓ 	/␈ε(U␈↓ 	m␈ε"<␈↓ 
≠␈ε6↓␈αε↓␈αε↓␈↓ 
K␈ε".
␈βu␈↓ β␈ε+n␈↓ πp␈ε+s␈↓ λ\␈ε+s␈↓ 	H␈ε+s
␈β␈␈↓ π}␈ε'0␈↓ λj␈ε'1␈↓ 	V␈ε'2
␈β∀␈↓ α␈ε"The␈α
secret␈α∞is␈α
to␈α
restrict␈α
the␈α∞subsequences␈α
so␈α
that␈α
they␈α∞could␈α
be␈α
de|ned␈α
by
␈β?␈↓ ↓H␈ε"somebody␈αwh␈α↓o␈αdoes␈αn␈α↓ot␈αlo␈α↓ok␈αat␈↓ ¬F␈ε(U␈↓ ¬⎇␈ε"before␈αdeciding␈αwhether␈αor␈αn␈α↓ot␈αit␈αis␈αto␈αbe␈αin
␈βK␈↓ ¬↑␈ε+n
␈βj␈↓ ↓H␈ε"the␈αsubsequence.␈α⊂The␈αfollo␈α␈wing␈αde|nition␈αn␈α↓o␈α␈w␈αsuggests␈αitself:
␈β
(␈↓ ↓H␈ε2De|nition␈α	R4.␈ε/␈α∂A␈↓ βW␈ε"[␈↓ βh␈ε"0,␈αε1)␈ε/␈α	sequence␈ε6␈α	h␈↓ ¬N␈ε(U␈↓ ¬z␈ε6i␈ε/␈α	is␈α	said␈α	to␈α
be␈α	\random"␈α	if,␈α
for␈α	ev␈α␈ery␈α	e{ectiv␈α␈e
␈β
5␈↓ ¬g␈ε+n
␈β
S␈↓ ↓H␈ε/algorithm␈α
that␈α
speci|es␈α
an␈α
in|nite␈α
sequence␈α
of␈α
distinct␈α∞n␈α↓onnegativ␈α␈e␈α
in␈α␈tegers␈↓ 	␈ε(s
␈β
`␈↓ ~␈ε+n
␈β
␈␈↓ ↓H␈ε/for␈ε(␈αn␈ε6␈α∃␈ε"␈α
0␈ε/,␈α
the␈αsubsequence␈↓ ¬↓␈ε(U␈↓ ¬5␈ε/,␈↓ ¬K␈ε(U␈↓ ¬␈␈ε/,␈↓ ε⊗␈ε(U␈↓ εJ␈ε/,␈↓ ε`␈ε".␈αε.␈αε.␈↓ π↔␈ε/corresponding␈αto␈α
this␈αalgorithm␈αis
␈β∞␈↓ ¬→␈ε+s␈↓ ¬d␈ε+s␈↓ ε.␈ε+s
␈β∞∃␈↓ ¬'␈ε'0␈↓ ¬r␈ε'1␈↓ ε<␈ε'2
␈β∞*␈↓ ↓H␈ε61␈ε/-distributed.
␈β∞h␈↓ α␈ε"The␈α⊂algorithms␈α⊃referred␈α⊂to␈α⊂in␈α⊂De|nition␈α⊂R4␈↓ πd␈ε"are␈α⊂e{ectiv␈α␈e␈α⊂procedures␈α⊂that
␈β∂∪␈↓ ↓H␈ε"compute␈↓ αa␈ε(s␈↓ β∧␈ε",␈α∃giv␈α␈en␈ε(␈α∀n␈ε"␈α␈.␈α'(See␈α∪Section␈α∪1.1.)␈α&Th␈α␈us,␈α∃for␈α∪example,␈α∃the␈α∪sequence
␈β∂∨␈↓ αr␈ε+n
␈β∂8␈↓ ↓j␈ε+n
␈β∂>␈↓ ↓H␈ε6h␈↓ ↓T␈ε(→␈↓ αα␈ε"mod␈↓ αL␈ε"1␈ε6i␈ε"␈αwill␈ε/␈α
n␈α↓ot␈ε"␈αsatisfy␈α
R4,␈αsince␈α
it␈αis␈α
either␈αn␈α↓ot␈α
equidistributed␈αor␈α
there␈αis␈α
an
␈β∂c␈↓ 	u␈ε+s
␈β∂i␈↓ ↓H␈ε"e{ectiv␈α␈e␈αλalgorithm␈α	that␈α	determines␈↓ ¬l␈ε"an␈α	in|nite␈α	subsequence␈↓ λX␈ε(s␈↓ 	¬␈ε"with␈αλ(␈↓ 	←␈ε(→␈↓ 
⊗␈ε"mod␈↓ 
`␈ε"1)␈α
<
␈β∂m␈↓ 
β␈ε'0
␈β∂v␈↓ λi␈ε+n
␈β⊂∂␈↓ ↓j␈ε+s␈↓ βS␈ε+s
␈β⊂∃␈↓ ↓H␈ε"(␈↓ ↓T␈ε(→␈↓ α␈ε"mod␈↓ αU␈ε"1)␈α
<␈α
(␈↓ β=␈ε(→␈↓ βt␈ε"mod␈↓ ∧>␈ε"1)␈α
<␈↓ ¬~␈ε6↓␈αε↓␈αε↓␈↓ ¬J␈ε".␈α⊗Similarly,␈ε/␈α∞n␈α↓o␈α∞explicitly␈α∞de|ned␈α∞sequence␈α
can
␈β⊂_␈↓ ↓x␈ε'1␈↓ βa␈ε'2
␈β⊂@␈↓ ↓H␈ε/satisfy␈α
De|nition␈α
R4␈↓ ∧∪␈ε";␈α∞this␈α
is␈α
appropriate,␈α
if␈α
w␈α␈e␈α
agree␈α
that␈α
n␈α↓o␈α
explicitly␈αde|ned
␈β⊂k␈↓ ↓H␈ε"sequence␈α∂can␈α⊂really␈α∂be␈α⊂random.␈α~It␈α⊂is␈α∂quite␈α⊂lik␈α␈ely,␈α⊂h␈α↓o␈α␈w␈α␈ev␈α␈er,␈α⊂that␈α⊂the␈α∂sequence
␈β⊃⊂␈↓ ↓e␈ε+n
␈β⊃⊗␈↓ ↓H␈ε6h␈↓ ↓T␈ε(∩␈↓ ↓⎇␈ε"mod␈↓ αG␈ε"1␈ε6i␈ε"␈α
will␈α
satisfy␈α	De|nition␈α
R4,␈α
for␈α
alm␈α↓ost␈α
all␈α	real␈α
n␈α␈um␈α␈bers␈↓ 	0␈ε(∩␈↓ 	L␈ε">␈α
1;␈α
this␈α
is␈α	n␈α↓o
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"155
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα$␈↓ ↓H␈ε"con␈α␈tradiction,␈αsince␈α
alm␈α↓ost␈αall␈↓ ¬&␈ε(∩␈↓ ¬B␈ε"are␈αuncomputable␈αby␈α
algorithms.␈α⊂The␈α
follo␈α␈wing
␈βαI␈↓ πK␈ε+n
␈βαO␈↓ ↓H␈ε"facts␈α
are␈α
kn␈α↓o␈α␈wn,␈αfor␈α
example:␈α∩(i)␈αThe␈α
sequence␈ε6␈α
h␈↓ π:␈ε(∩␈↓ πd␈ε"mod␈↓ λ.␈ε"1␈ε6i␈ε"␈α
satis|es␈α
De|nition␈α
R4
␈βαz␈↓ ↓H␈ε"for␈α
alm␈α↓ost␈α
all␈α
real␈↓ βe␈ε(∩␈↓ ∧↓␈ε">␈α
1,␈α
if␈α
\␈ε61␈ε"-distributed"␈αis␈α
replaced␈α
by␈α
\1-distributed."␈α∂This
␈ββ&␈↓ ↓H␈ε"theorem␈α
w␈α␈as␈α
pro␈α␈v␈α␈ed␈α
by␈α
J.␈α
F.␈α
Koksma,␈ε/␈α
Compositio␈α
Mathematica␈ε2␈α
2␈ε"␈α
(1935),␈α
250↑
␈ββK␈↓ ε
␈ε+s␈ε%(␈ε+␈α↓n␈ε%␈α␈)
␈ββQ␈↓ ↓H␈ε"258.␈α∨(ii)␈α∂The␈α∂particular␈α⊂sequence␈ε6␈α∂h␈↓ ¬|␈ε(∩␈↓ εI␈ε"mod␈↓ π∪␈ε"1␈ε6i␈ε"␈α∂is␈ε6␈α∂1␈ε"-distributed␈α⊂for␈α∂alm␈α↓ost␈α∂all
␈ββ|␈↓ ↓H␈ε"real␈↓ α∞␈ε(∩␈↓ α)␈ε">␈α1,␈αif␈ε6␈αh␈ε(s␈ε"(␈ε(n␈ε")␈ε6i␈ε"␈αis␈αa␈α
sequence␈αof␈αin␈α␈tegers␈αfor␈αwhich␈ε(␈α
s␈ε"(␈ε(n␈ε"␈απ+␈α	1)␈ε6␈αλ␈␈ε(␈αλs␈ε"(␈ε(n␈ε")␈ε6␈α
!␈α
1␈ε"␈αas
␈β∧!␈↓ π!␈ε%2
␈β∧'␈↓ ↓H␈ε(n␈ε6␈α	!␈α
1␈ε".␈α⊂F␈α⎇or␈αexample,␈αw␈α␈e␈αcould␈αha␈α␈v␈α␈e␈ε(␈αs␈ε"(␈ε(n␈ε")␈α
=␈↓ π␈ε(n␈↓ π0␈ε",␈αor␈ε(␈αs␈ε"(␈ε(n␈ε")␈α
=␈ε6␈α
b␈ε(n␈↓ 	∪␈ε"lg␈↓ 	5␈ε(n␈ε6␈α␈c␈ε".
␈β∧T␈↓ α␈ε"De|nition␈αR4␈αis␈αm␈α␈uch␈αstronger␈αthan␈αDe|nition␈αR1;␈αbut␈αit␈αis␈αstill␈αreasonable
␈β∧␈␈↓ ↓H␈ε"to␈α	claim␈α
that␈α
De|nition␈α
R4␈α
is␈α
to␈α↓o␈α
w␈α␈eak.␈α∂F␈α⎇or␈α
example,␈α
let␈ε6␈α
h␈↓ λS␈ε(U␈↓ λ}␈ε6i␈ε"␈α
be␈α
a␈α
truly␈α	random
␈β¬␈↓ λk␈ε+n
␈β¬+␈↓ ↓H␈ε"sequence,␈α
and␈α
de|ne␈α
the␈α
subsequence␈ε6␈α∞h␈↓ ε.␈ε(U␈↓ εd␈ε6i␈ε"␈α
by␈α∞the␈α
follo␈α␈wing␈α
rules:␈↓ 	m␈ε(s␈↓ 
→␈ε"=␈α0,␈α
and
␈β¬7␈↓ εF␈ε+s␈↓ 	}␈ε%0
␈β¬A␈↓ εT␈ε-n
␈β¬V␈↓ ↓H␈ε"(for␈ε(␈αn␈ε"␈α
>␈α0)␈↓ β¬␈ε(s␈↓ β5␈ε"is␈αthe␈α
smallest␈αin␈α␈teger␈ε6␈α
∃␈ε(␈α
n␈ε"␈αfor␈αwhich␈↓ λ␈ε(U␈↓ λn␈ε",␈↓ 	¬␈ε(U␈↓ 	i␈ε",␈↓ 	␈␈ε".␈αε.␈αε.␈↓ 
/␈ε",␈↓ 
E␈ε(U
␈β¬b␈↓ β⊗␈ε+n␈↓ λ$␈ε+s␈↓ λB␈ε9␈␈ε%␈α␈1␈↓ 	≡␈ε+s␈↓ 	<␈ε9␈␈ε%␈α␈2␈↓ 
↑␈ε+s␈↓ 
|␈ε9␈␈ε+n
␈β¬l␈↓ λ2␈ε-n␈↓ 	,␈ε-n␈↓ 
l␈ε-n
␈β¬|␈↓ βO␈ε%1
␈βε↓␈↓ ↓H␈ε"are␈αall␈αless␈αthan␈↓ βb␈ε".␈α⊂Th␈α␈us␈αw␈α␈e␈αare␈αconsidering␈αthe␈αsubsequence␈αof␈αv␈α⎇alues␈αfollo␈α␈wing
␈βε∩␈↓ βO␈ε%2
␈βε∃␈↓ βO␈∧ε∃βOα⊂
␈βε(␈↓ π/␈ε%1␈↓ 
>␈ε%1
␈βε,␈↓ ↓H␈ε"the␈α
|rst␈α∞consecutiv␈α␈e␈α∞run␈α
of␈ε(␈α∞n␈ε"␈α∞v␈α⎇alues␈α
less␈α∞than␈↓ πB␈ε".␈α∃Suppose␈α∞that␈α
\␈↓ 	P␈ε(U␈↓ 
	␈ε"<␈↓ 
Q␈ε""␈α
cor-
␈βε9␈↓ 	i␈ε+n
␈βε=␈↓ π/␈ε%2␈↓ 
>␈ε%2
␈βε@␈↓ π/␈∧ε@π/α⊂␈↓ 
>␈∧ε@
>α⊂
␈βεW␈↓ ↓H␈ε"responds␈α∞to␈α∞the␈α∞v␈α⎇alue␈α∞\heads"␈α∞in␈α∞the␈α∞⎇ipping␈α∂of␈α∞a␈α∞coin.␈α⊗Gam␈α␈blers␈α∞tend␈α∞to␈α∞feel
␈βπβ␈↓ ↓H␈ε"that␈α
a␈α
long␈αrun␈α
of␈α
\heads"␈αmak␈α␈es␈α
the␈α
opposite␈αcondition,␈α
\tails,"␈αm␈α↓ore␈α
probable,
␈βπ.␈↓ ↓H␈ε"assuming␈αthat␈αa␈α
true␈αcoin␈αis␈αbeing␈α
used;␈αand␈αthe␈α
subsequence␈ε6␈αh␈↓ 	%␈ε(U␈↓ 	\␈ε6i␈ε"␈αjust␈αde|ned
␈βπ:␈↓ 	>␈ε+s
␈βπD␈↓ 	L␈ε-n
␈βπY␈↓ ↓H␈ε"corresponds␈αto␈αa␈↓ βU␈ε"gam␈α␈bling␈αsystem␈αfor␈αa␈αman␈αwh␈α↓o␈αplaces␈αhis␈ε(␈αn␈ε"th␈αbet␈αon␈αthe␈αcoin
␈βλ∧␈↓ ↓H␈ε"toss␈α∞follo␈α␈wing␈α∂the␈α∂|rst␈α∞run␈α∂of␈ε(␈α∂n␈ε"␈α∞consecutiv␈α␈e␈α∂\heads."␈α_The␈α∞gam␈α␈bler␈α∂ma␈α␈y␈α∞think
␈βλ+␈↓ β9␈ε%1␈↓ ¬3␈ε%1
␈βλ/␈↓ ↓H␈ε"that␈↓ α∀␈ε"P␈↓ α,␈ε"r␈↓ α:␈ε"(␈↓ αF␈ε(U␈↓ βπ␈ε6∃␈↓ βL␈ε")␈α
is␈α
m␈α↓ore␈α
than␈↓ ¬F␈ε",␈αbut␈α
of␈α
course␈α
in␈α
a␈α
truly␈α
random␈α
sequence␈ε6␈α
h␈↓ 
j␈ε(U␈↓  ␈ε6i
␈βλ<␈↓ α←␈ε+s␈↓ α␈ε+s
␈βλ@␈↓ β9␈ε%2␈↓ ¬3␈ε%2
␈βλD␈↓ β9␈∧λDβ9α⊂␈↓ ¬3␈∧λD¬3α⊂
␈βλE␈↓ αm␈ε-n␈↓ ⊂␈ε-n
␈βλ[␈↓ ↓H␈ε"will␈α∞be␈α∞completely␈α∂random.␈α↔No␈α∞gam␈α␈bling␈α∂system␈α∞will␈α∂ev␈α␈er␈α∞be␈α∞able␈α∂to␈α∞beat␈α∞the
␈β	ε␈↓ ↓H␈ε"odds!␈α⊃De|nition␈αR4␈αsa␈α␈ys␈α
n␈α↓othing␈αabout␈αsubsequences␈αformed␈α
according␈αto␈αsuch
␈β	1␈↓ ↓H␈ε"a␈αgam␈α␈bling␈αsystem,␈αso␈αapparen␈α␈tly␈αw␈α␈e␈αneed␈αsomething␈αm␈α↓ore.
␈β	↑␈↓ α␈ε"Let␈α∂us␈α∂de|ne␈α⊂a␈α∂\␈↓ ∧!␈ε"subsequence␈α∂rule"␈ε6␈α∂R␈ε"␈α∂as␈α⊂an␈α∂in|nite␈α∂sequence␈α∂of␈α∂functions
␈β
	␈↓ ↓H␈ε6h␈↓ ↓T␈ε(f␈↓ ↓x␈ε"(␈↓ α∧␈ε(x␈↓ α'␈ε",␈↓ α7␈ε".␈αε.␈αε.␈↓ αg␈ε",␈↓ αw␈ε(x␈↓ β≡␈ε")␈ε6i␈ε"␈α∩where,␈α∀for␈ε(␈α∪n␈ε6␈α∃∃␈ε"␈α∀0,␈↓ ε∃␈ε(f␈↓ εK␈ε"is␈α∪a␈α∩function␈α∪of␈ε(␈α∪n␈ε"␈α∩v␈α⎇ariables,␈α∀and␈α∩the
␈β
⊗␈↓ ↓e␈ε+n␈↓ α_␈ε%1␈↓ β␈ε+n␈↓ ε&␈ε+n
␈β
4␈↓ ↓H␈ε"v␈α⎇alue␈α∞of␈↓ αS␈ε(f␈↓ αw␈ε"(␈↓ ββ␈ε(x␈↓ β'␈ε",␈↓ β7␈ε".␈αε.␈αε.␈↓ βg␈ε",␈↓ βw␈ε(x␈↓ ∧≥␈ε")␈α∂is␈α∂either␈α∞0␈α∂or␈α∂1.␈α_Here␈↓ π"␈ε(x␈↓ πE␈ε",␈↓ π↑␈ε".␈αε.␈αε.␈↓ λ∞␈ε",␈↓ λ(␈ε(x␈↓ λ]␈ε"are␈α∂elemen␈α␈ts␈α∂of␈α∞some
␈β
A␈↓ αe␈ε+n␈↓ β↔␈ε%1␈↓ ∧␈ε+n␈↓ π6␈ε%1␈↓ λ<␈ε+n
␈β
`␈↓ ↓H␈ε"set␈↓ αβ␈ε(S␈↓ α≤␈ε".␈α≡(Th␈α␈us,␈α⊂in␈α∂particular,␈↓ ¬~␈ε(f␈↓ ¬J␈ε"is␈α∂a␈α∂constan␈α␈t␈α∂function,␈α⊂either␈α∂0␈α∂or␈α∂1.)␈α≡A␈α∂sub-
␈β
l␈↓ ¬,␈ε%0
␈β␈↓ ↓H␈ε"sequence␈αrule␈ε6␈αR␈ε"␈αde|nes␈αa␈αsubsequence␈αof␈αan␈α␈y␈αin|nite␈α
sequence␈ε6␈αh␈↓ 	3␈ε(X␈↓ 	c␈ε6i␈ε"␈αof␈αelemen␈α␈ts
␈β↔␈↓ 	Q␈ε+n
␈β6␈↓ ↓H␈ε"of␈↓ ↓u␈ε(S␈↓ α≥␈ε"as␈α∂follo␈α␈ws:␈ε/␈α∩The␈ε(␈α∂n␈ε/th␈α∂term␈↓ ¬@␈ε(X␈↓ ε␈ε/is␈α∂in␈α∂the␈α∂subsequence␈ε6␈α∂h␈↓ λn␈ε(X␈↓ 	∨␈ε6iR␈ε/␈α∂if␈α∂and␈α∂only␈α∂if
␈βB␈↓ ¬↑␈ε+n␈↓ 	␈ε+n
␈βa␈↓ ↓H␈ε(f␈↓ ↓l␈ε"(␈↓ ↓x␈ε(X␈↓ α%␈ε",␈↓ α5␈ε(X␈↓ αb␈ε",␈↓ αr␈ε".␈αε.␈αε.␈↓ β"␈ε",␈↓ β2␈ε(X␈↓ ∧⊂␈ε")␈α∂=␈α⊂1␈ε/.␈ε"␈α→Note␈α∂that␈α⊂the␈α∂subsequence␈ε6␈α∂h␈↓ λ[␈ε(X␈↓ 	␈ε6iR␈ε"␈α∂th␈α␈us␈α∂de|ned␈α∂is
␈βn␈↓ ↓Y␈ε+n␈↓ α⊗␈ε%0␈↓ αS␈ε%1␈↓ βP␈ε+n␈ε9␈␈ε%1␈↓ λy␈ε+n
␈β␈↓ ↓H␈ε"n␈α↓ot␈αnecessarily␈αin|nite,␈αand␈αit␈αma␈α␈y␈αin␈αfact␈αcon␈α␈tain␈αn␈α↓o␈αelemen␈α␈ts␈αat␈αall.
␈β9␈↓ α␈ε"F␈α⎇or␈α∞example,␈α∂the␈α∂gam␈α␈bler's␈α∞subsequence␈α∂just␈α∞described␈α∂corresponds␈α∞to␈α∞the
␈βe␈↓ ↓H␈ε"follo␈α␈wing␈α
subsequence␈α∞rule:␈α∂\␈↓ ¬∃␈ε(f␈↓ ¬C␈ε"=␈α1;␈α∂and␈α
for␈ε(␈α∞n␈ε"␈α>␈α
0,␈↓ λ≤␈ε(f␈↓ λ@␈ε"(␈↓ λL␈ε(x␈↓ λo␈ε",␈↓ λ␈␈ε".␈αε.␈αε.␈↓ 	/␈ε",␈↓ 	?␈ε(x␈↓ 	f␈ε")␈α
=␈α1␈α∞if␈α
and
␈βq␈↓ ¬'␈ε%0␈↓ λ.␈ε+n␈↓ λ`␈ε%1␈↓ 	S␈ε+n
␈β
⊂␈↓ ↓H␈ε"only␈α∂if␈α⊂there␈α∂is␈α∂some␈↓ ∧$␈ε(k␈↓ ∧H␈ε"in␈α∂the␈α⊂range␈α∂0␈α⊂<␈↓ εr␈ε(k␈↓ π∃␈ε6∀␈ε(␈α⊂n␈ε"␈α∂such␈α⊂that␈α∂the␈↓ 	W␈ε(k␈↓ 	z␈ε"consecutiv␈α␈e
␈β
6␈↓ πK␈ε%1
␈β
;␈↓ ↓H␈ε"parameters␈↓ β∧␈ε(x␈↓ β3␈ε",␈↓ βL␈ε(x␈↓ ∧'␈ε",␈↓ ∧A␈ε".␈αε.␈αε.␈↓ ∧q␈ε",␈↓ ¬
␈ε(x␈↓ ε"␈ε"are␈α∂all␈α∞<␈↓ πm␈ε"when␈ε(␈α∂m␈ε"␈α∞=␈ε(␈α∞n␈ε"␈α∞but␈α∂n␈α↓ot␈α∞when
␈β
G␈↓ β_␈ε+m␈↓ β`␈ε+m␈ε9␈␈ε%␈α␈1␈↓ ¬≡␈ε+m␈ε9␈α␈␈␈↓ ¬V␈ε+k␈↓ ¬g␈ε%+␈α␈1
␈β
L␈↓ πK␈ε%2
␈β
O␈↓ πK␈∧
OπKα⊂
␈β
f␈↓ ↓H␈ε(k␈↓ ↓f␈ε6∀␈ε(␈α
m␈ε"␈α
<␈ε(␈α
n␈ε"␈α␈."
␈β∞∪␈↓ α␈ε"A␈α
subsequence␈α
rule␈ε6␈αR␈ε"␈α
is␈α
said␈αto␈α
be␈ε/␈α
computable␈ε"␈α
if␈αthere␈↓ λ}␈ε"is␈α
an␈α
e{ectiv␈α␈e␈αalgo-
␈β∞>␈↓ ↓H␈ε"rithm␈α∞that␈α∂determines␈α∂the␈α∞v␈α⎇alue␈α∂of␈↓ ε∧␈ε(f␈↓ ε(␈ε"(␈↓ ε4␈ε(x␈↓ εW␈ε",␈↓ εg␈ε".␈αε.␈αε.␈↓ π↔␈ε",␈↓ π'␈ε(x␈↓ πN␈ε"),␈α∂when␈ε(␈α∂n␈ε"␈α∞and␈↓ 	A␈ε(x␈↓ 	d␈ε",␈↓ 	}␈ε".␈αε.␈αε.␈↓ 
.␈ε",␈↓ 
G␈ε(x␈↓ 
|␈ε"are
␈β∞K␈↓ ε⊗␈ε+n␈↓ εH␈ε%1␈↓ π;␈ε+n␈↓ 	U␈ε%1␈↓ 
[␈ε+n
␈β∞j␈↓ ↓H␈ε"giv␈α␈en␈α	as␈α
input.␈α∂We␈α
had␈α	better␈α
restrict␈α	ourselv␈α␈es␈α
to␈α	computable␈α
subsequence␈α	rules
␈β∂∃␈↓ ↓H␈ε"when␈αtrying␈αto␈αde|ne␈αrandomness,␈αlest␈αw␈α␈e␈αobtain␈αan␈αo␈α␈v␈α␈erly␈αrestrictiv␈α␈e␈αde|nition
␈β∂@␈↓ ↓H␈ε"lik␈α␈e␈α∞R3␈α∂abo␈α␈v␈α␈e.␈α_But␈α∂e{ectiv␈α␈e␈α∂algorithms␈α∂cann␈α↓ot␈α∞deal␈α∂nicely␈α∂with␈α∂arbitrary␈α∞real
␈β∂k␈↓ ↓H␈ε"n␈α␈um␈α␈bers␈α∂as␈α∂inputs;␈α⊃for␈α⊂example,␈α⊂if␈α∂a␈α⊂real␈α∂n␈α␈um␈α␈ber␈ε(␈α⊂x␈ε"␈α∂is␈α∂speci|ed␈α⊂by␈α∂an␈α∂in|nite
␈β⊂∩␈↓ 	J␈ε%1
␈β⊂⊗␈↓ ↓H␈ε"radix-10␈αexpansion,␈αthere␈αis␈αn␈α↓o␈αalgorithm␈αto␈αdetermine␈αif␈ε(␈αx␈ε"␈αis␈α<␈↓ 	i␈ε"or␈αn␈α↓ot,␈αsince
␈β⊂'␈↓ 	J␈ε%3
␈β⊂+␈↓ 	J␈∧⊂+	Jα⊂
␈β⊂B␈↓ ↓H␈ε"all␈α∞digits␈α∂of␈α∞the␈α∞n␈α␈um␈α␈ber␈α∂0.333␈↓ ¬*␈ε".␈αε.␈αε.␈↓ ¬b␈ε"ha␈α␈v␈α␈e␈α∂to␈α∞be␈α∂examined.␈α↔Therefore␈α∞computable
␈β⊂m␈↓ ↓H␈ε"subsequence␈α∂rules␈α∂do␈α∂n␈α↓ot␈α∂apply␈α∂to␈α∂all␈↓ ε1␈ε"[␈↓ εA␈ε"0,␈αε1)␈α∂sequences,␈α⊂and␈α∂it␈α∂is␈α∂con␈α␈v␈α␈enien␈α␈t␈α∂to
␈β⊃_␈↓ ↓H␈ε"base␈αour␈αnext␈αde|nition␈αon␈ε(␈αb␈ε"-ary␈αsequences:
␈β∪(

␈β↓U␈↓ ↓H␈ε"156␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα$␈↓ ↓H␈ε2De|nition␈α∞R5.␈ε/␈α∨A␈ε(␈α∂b␈ε/-ary␈α∂sequence␈α∂is␈α∞said␈α∂to␈α∂be␈α∂\random"␈α∂if␈α∂ev␈α␈ery␈α∂in|nite␈α∞sub-
␈βαO␈↓ ↓H␈ε/sequence␈αde|ned␈αby␈αa␈αcomputable␈αsubsequence␈αrule␈αis␈ε"␈α1␈ε/-distributed.
␈ββ¬␈↓ α␈ε/A␈↓ α3␈ε"[␈↓ αC␈ε"0,␈αε1)␈ε/␈α
sequence␈ε6␈α
h␈↓ ∧1␈ε(U␈↓ ∧\␈ε6i␈ε/␈α
is␈α
said␈αto␈α
be␈α
\random"␈α
if␈αthe␈ε(␈α
b␈ε/-ary␈α
sequence␈ε6␈α
hb␈ε(b␈↓ 
g␈ε(U␈↓ ∩␈ε6ci
␈ββ⊃␈↓ ∧I␈ε+n␈↓ ␈ε+n
␈ββ0␈↓ ↓H␈ε/is␈α\random"␈αfor␈αall␈αin␈α␈tegers␈ε(␈αb␈ε6␈α
∃␈ε"␈α
2␈ε/.
␈ββp␈↓ α␈ε"Note␈αthat␈α
De|nition␈αR5␈α
sa␈α␈ys␈αonly␈α\1-distributed,"␈α
n␈α↓ot␈α\␈ε61␈ε"-distributed."␈α⊃It
␈β∧≠␈↓ ↓H␈ε"is␈α∂in␈α␈teresting␈α⊂to␈α⊂v␈α␈erify␈α⊂that␈α⊂this␈α∂ma␈α␈y␈α⊂be␈α⊂done␈α⊂with␈α↓out␈α⊂loss␈α∂of␈α⊂generalit␈α␈y.␈α≠F␈α⎇or
␈β∧F␈↓ ↓H␈ε"w␈α␈e␈αma␈α␈y␈αde|ne␈αan␈αobviously␈αcomputable␈αsubsequence␈αrule␈ε6␈αR␈ε"(␈↓ λy␈ε(a␈↓ 	"␈ε".␈αε.␈αε.␈↓ 	R␈ε(a␈↓ 	u␈ε")␈αas␈αfollo␈α␈ws,
␈β∧S␈↓ 	␈ε%1␈↓ 	e␈ε+k
␈β∧q␈↓ ↓H␈ε"giv␈α␈en␈α
an␈α␈y␈ε(␈α
b␈ε"-ary␈α
n␈α␈um␈α␈ber␈↓ ∧?␈ε(a␈↓ ∧h␈ε".␈αε.␈αε.␈↓ ¬_␈ε(a␈↓ ¬;␈ε":␈αLet␈↓ ε⊂␈ε(f␈↓ ε5␈ε"(␈↓ εA␈ε(x␈↓ εd␈ε",␈↓ εt␈ε".␈αε.␈αε.␈↓ π$␈ε",␈↓ π4␈ε(x␈↓ πZ␈ε")␈α
=␈α
1␈αif␈α
and␈α
only␈α
if␈ε(␈α
n␈ε6␈α
∃␈↓ 
Z␈ε(k␈↓ 
r␈ε6␈␈ε"␈α∧1
␈β∧}␈↓ ∧R␈ε%1␈↓ ¬+␈ε+k␈↓ ε"␈ε+n␈↓ εU␈ε%1␈↓ πH␈ε+n
␈β¬≥␈↓ ↓H␈ε"and␈↓ α∞␈ε(x␈↓ β→␈ε"=␈↓ βG␈ε(a␈↓ βj␈ε",␈↓ ∧␈ε".␈αε.␈αε.␈↓ ∧0␈ε",␈↓ ∧F␈ε(x␈↓ ¬#␈ε"=␈↓ ¬Q␈ε(a␈↓ ε"␈ε",␈↓ ε8␈ε(x␈↓ εi␈ε"=␈↓ π↔␈ε(a␈↓ π:␈ε".␈α⊂No␈α␈w␈αif␈ε6␈α
h␈↓ λV␈ε(X␈↓ 	ε␈ε6i␈ε"␈αis␈αa␈↓ 	`␈ε(k␈↓ 	t␈ε"-distributed
␈β¬)␈↓ α"␈ε+n␈ε9␈α␈␈␈↓ αR␈ε+k␈↓ αb␈ε%+1␈↓ βZ␈ε%1␈↓ ∧Z␈ε+n␈ε9␈α␈␈␈ε%1␈↓ ¬d␈ε+k␈↓ ¬u␈ε9␈␈ε%1␈↓ εL␈ε+n␈↓ π*␈ε+k␈↓ λt␈ε+n
␈β¬H␈↓ ↓H␈ε(b␈ε"-ary␈α∞sequence,␈α∞this␈α∞rule␈ε6␈α∞R␈ε"(␈↓ ∧␈␈ε(a␈↓ ¬'␈ε".␈αε.␈αε.␈↓ ¬W␈ε(a␈↓ ¬{␈ε")←which␈α∞selects␈α∞the␈α∞subsequence␈α
consisting
␈β¬T␈↓ ¬∩␈ε%1␈↓ ¬j␈ε+k
␈β¬s␈↓ ↓H␈ε"of␈αth␈α↓ose␈α
terms␈αjust␈αfollo␈α␈wing␈α
an␈αoccurrence␈α
of␈↓ π'␈ε(a␈↓ πO␈ε".␈αε.␈αε.␈↓ π␈␈ε(a␈↓ λ#␈ε"←de|nes␈α
an␈αin|nite␈αsub-
␈β¬␈␈↓ π:␈ε%1␈↓ λ∩␈ε+k
␈βε≡␈↓ ↓H␈ε"sequence;␈α∩and␈α⊃if␈α⊂this␈α⊂subsequence␈α⊃is␈α⊂1-distributed,␈α⊃each␈α⊃of␈α⊂the␈α⊂(␈↓ 	V␈ε(k␈↓ 	v␈ε"+␈α
1)-tuples
␈βεC␈↓ 	α␈ε+k␈↓ 	∩␈ε%+1
␈βεI␈↓ ↓H␈ε(a␈↓ ↓p␈ε".␈αε.␈αε.␈↓ α ␈ε(a␈↓ αC␈ε(a␈↓ β!␈ε"for␈α
0␈ε6␈α∀␈↓ ∧(␈ε(a␈↓ ¬∧␈ε"<␈ε(␈αb␈ε"␈α∞occurs␈α
with␈α
probabilit␈α␈y␈α
1/␈↓ λq␈ε(b␈↓ 	L␈ε"in␈ε6␈α
h␈↓ 
β␈ε(X␈↓ 
4␈ε6i␈ε".␈α∪Th␈α␈us
␈βεV␈↓ ↓[␈ε%1␈↓ α3␈ε+k␈↓ αV␈ε+k␈↓ αg␈ε%+1␈↓ ∧;␈ε+k␈↓ ∧L␈ε%+␈α␈1␈↓ 
!␈ε+n
␈βεu␈↓ ↓H␈ε"w␈α␈e␈αcan␈αpro␈α␈v␈α␈e␈α
that␈αa␈α
sequence␈αsatisfying␈α
De|nition␈αR5␈αis␈↓ λK␈ε(k␈↓ λ←␈ε"-distributed␈αfor␈α
all␈↓ ∞␈ε(k␈↓ "␈ε",
␈βπ ␈↓ ↓H␈ε"by␈α
induction␈α
on␈↓ βP␈ε(k␈↓ βd␈ε".␈α∪Similarly,␈α∞by␈α
considering␈α
the␈α
\composition"␈α
of␈α
subsequence
␈βπK␈↓ ↓H␈ε"rules←if␈↓ αW␈ε6R␈↓ β
␈ε"de|nes␈αan␈αin|nite␈αsubsequence␈ε6␈αh␈↓ π↓␈ε(X␈↓ π2␈ε6i␈↓ π>␈ε6R␈↓ πi␈ε",␈αthen␈αw␈α␈e␈αcan␈αde|ne␈↓ 
+␈ε6R␈↓ 
V␈ε6R␈↓ ␈ε"to
␈βπW␈↓ αs␈ε%1␈↓ π∨␈ε+n␈↓ πZ␈ε%1␈↓ 
G␈ε%1␈↓ 
r␈ε%2
␈βπv␈↓ ↓H␈ε"be␈α∞the␈α∂subsequence␈α∂rule␈α∂for␈α∞which␈ε6␈α∂h␈↓ ε∧␈ε(X␈↓ ε5␈ε6i␈↓ εA␈ε6R␈↓ εl␈ε6R␈↓ π&␈ε"=␈α∞(␈ε6h␈↓ πp␈ε(X␈↓ λ!␈ε6i␈↓ λ-␈ε6R␈↓ λX␈ε")␈↓ λd␈ε6R␈↓ 	∂␈ε"←w␈α␈e␈α∂|nd␈α∂that␈α∞all
␈βλβ␈↓ ε"␈ε+n␈↓ ε]␈ε%1␈↓ πλ␈ε%2␈↓ λ∞␈ε+n␈↓ λI␈ε%1␈↓ 	␈ε%2
␈βλ!␈↓ ↓H␈ε"subsequences␈αconsidered␈αin␈αDe|nition␈αR5␈αare␈ε6␈α1␈ε"-distributed.␈α→(See␈αexercise␈α32.)
␈βλM␈↓ α␈ε"The␈α∞fact␈α∞that␈ε6␈α
1␈ε"-distribution␈α∞comes␈α∞out␈α∞of␈α∞De|nition␈α
R5␈α∞as␈α∞a␈α∞v␈α␈ery␈α
special
␈βλx␈↓ ↓H␈ε"case␈αλis␈α	encouraging,␈α	and␈α	it␈α	is␈αλa␈α	go␈α↓od␈α	indication␈α	that␈αλw␈α␈e␈α	ma␈α␈y␈α	at␈αλlast␈α	ha␈α␈v␈α␈e␈α	found␈αλthe
␈β	#␈↓ ↓H␈ε"de|nition␈α
of␈α
randomness␈α
w␈α␈e␈α
ha␈α␈v␈α␈e␈α
been␈α
seeking.␈α⊂But␈α
alas,␈α
there␈α
still␈α
is␈α
a␈α
problem.
␈β	N␈↓ ↓H␈ε"It␈α
is␈α	n␈α↓ot␈α
clear␈α
that␈α
sequences␈α
satisfying␈α
De|nition␈α
R5␈α
m␈α␈ust␈α
satisfy␈α
De|nition␈α	R4.
␈β	y␈↓ ↓H␈ε"The␈α∂\computable␈α∂subsequence␈α⊂rules"␈α∂w␈α␈e␈α⊂ha␈α␈v␈α␈e␈α∂just␈α⊂speci|ed␈α∂alw␈α␈a␈α␈ys␈α∂en␈α␈umerate
␈β
%␈↓ ↓H␈ε"subsequences␈ε6␈α⊃h␈↓ β1␈ε(X␈↓ βm␈ε6i␈ε"␈α⊃for␈α⊃which␈↓ ¬4␈ε(s␈↓ ¬f␈ε"<␈↓ ε≤␈ε(s␈↓ εO␈ε"<␈↓ π¬␈ε6↓␈αε↓␈αε↓␈↓ π5␈ε",␈α∩but␈ε6␈α⊃h␈↓ λ$␈ε(s␈↓ λH␈ε6i␈ε"␈α⊃does␈α⊃n␈α↓ot␈α⊃ha␈α␈v␈α␈e␈α⊂to␈α⊃be
␈β
1␈↓ βO␈ε+s␈↓ ¬E␈ε%0␈↓ ε-␈ε%1␈↓ λ5␈ε+n
␈β
;␈↓ β]␈ε-n
␈β
P␈↓ ↓H␈ε"m␈α↓on␈α↓otone␈αin␈αR4;␈αit␈αm␈α␈ust␈αonly␈αsatisfy␈αthe␈αcondition␈↓ πw␈ε(s␈↓ λ$␈ε6≤␈↓ λR␈ε(s␈↓ 	
␈ε"for␈ε(␈αn␈ε6␈α
≤␈ε(␈α
m␈ε"␈α␈.
␈β
\␈↓ λλ␈ε+n␈↓ λc␈ε+m
␈β
{␈↓ α␈ε"T␈α⎇o␈αmeet␈αthis␈αobjection,␈αw␈α␈e␈αma␈α␈y␈αcom␈α␈bine␈αDe|nitions␈αR4␈αand␈αR5␈αas␈αfollo␈α␈ws:
␈β;␈↓ ↓H␈ε2De|nition␈α	R6.␈ε/␈α∞A␈ε(␈α	b␈ε/-ary␈α	sequence␈ε6␈α	h␈↓ ¬K␈ε(X␈↓ ¬|␈ε6i␈ε/␈α	is␈α	said␈α	to␈α	be␈α	\random"␈α	if,␈α	for␈α	ev␈α␈ery␈α	e{ectiv␈α␈e
␈βG␈↓ ¬i␈ε+n
␈βf␈↓ ↓H␈ε/algorithm␈α
that␈αspeci|es␈αan␈αin|nite␈α
sequence␈αof␈αdistinct␈α
n␈α↓onnegativ␈α␈e␈αin␈α␈tegers␈ε6␈αh␈↓ 
⎇␈ε(s␈↓  ␈ε6i
␈βr␈↓ ∞␈ε+n
␈β⊃␈↓ ↓H␈ε/as␈α⊂a␈α⊂function␈α⊃of␈ε(␈α⊂n␈ε/␈α⊂and␈α⊂the␈α⊃v␈α⎇alues␈α⊂of␈↓ ε*␈ε(X␈↓ εd␈ε/,␈↓ ε␈␈ε".␈αε.␈αε.␈↓ π/␈ε/,␈↓ πJ␈ε(X␈↓ λ,␈ε/,␈α∩the␈α⊂subsequence␈ε6␈α⊂h␈↓ 
d␈ε(X␈↓  ␈ε6i
␈β≡␈↓ εH␈ε+s␈↓ πh␈ε+s␈↓ α␈ε+s
␈β'␈↓ εV␈ε'0␈↓ πv␈ε-n␈ε;␈α↓␈␈ε'1␈↓ ⊂␈ε-n
␈β<␈↓ ↓H␈ε/corresponding␈αto␈αthis␈αalgorithm␈αis␈α\random"␈αin␈αthe␈αsense␈αof␈αDe|nition␈αR5.
␈βr␈↓ α␈ε/A␈↓ α3␈ε"[␈↓ αC␈ε"0,␈αε1)␈ε/␈α
sequence␈ε6␈α
h␈↓ ∧1␈ε(U␈↓ ∧\␈ε6i␈ε/␈α
is␈α
said␈αto␈α
be␈α
\random"␈α
if␈αthe␈ε(␈α
b␈ε/-ary␈α
sequence␈ε6␈α
hb␈ε(b␈↓ 
g␈ε(U␈↓ ∩␈ε6ci
␈β}␈↓ ∧I␈ε+n␈↓ ␈ε+n
␈β
≥␈↓ ↓H␈ε/is␈α\random"␈αfor␈αall␈αin␈α␈tegers␈ε(␈αb␈ε6␈α
∃␈ε"␈α
2␈ε/.
␈β
]␈↓ α␈ε"The␈αauth␈α↓or␈αcon␈α␈tends*␈αthat␈αthis␈αde|nition␈αsurely␈αmeets␈αall␈αreasonable␈αphilo-
␈β∞λ␈↓ ↓H␈ε"sophical␈αrequiremen␈α␈ts␈αfor␈α
randomness,␈αso␈α
it␈αpro␈α␈vides␈α
an␈αansw␈α␈er␈α
to␈αthe␈αprincipal
␈β∞3␈↓ ↓H␈ε"question␈αposed␈αin␈αthis␈αsection.
␈β∞s␈↓ ↓H␈ε2D.␈αλExistence␈α	of␈αλrandom␈α	sequences.␈ε"␈αWe␈α	ha␈α␈v␈α␈e␈αλseen␈α	that␈αλDe|nition␈α	R3␈αλis␈α	to␈α↓o␈αλstrong,
␈β∂≡␈↓ ↓H␈ε"in␈αthe␈αsense␈αthat␈αn␈α↓o␈αsequence␈αcan␈αsatisfy␈αthat␈αde|nition;␈αand␈αthe␈αform␈α␈ulation␈αof
␈β∂I␈↓ ↓H␈ε"De|nitions␈α	R4,␈αR5,␈α
and␈α	R6␈α
abo␈α␈v␈α␈e␈α
w␈α␈as␈α
carried␈α	out␈α
in␈α
an␈α	attempt␈α
to␈α
recapture␈α	the
␈β∂u␈↓ ↓H␈ε"essen␈α␈tial␈αcharacteristics␈αof␈αDe|nition␈αR3.␈α⊂In␈αorder␈αto␈αsh␈α↓o␈α␈w␈αthat␈αDe|nition␈αR6␈αis
␈β⊂ ␈↓ ↓H␈ε"n␈α↓ot␈α	o␈α␈v␈α␈erly␈α
restrictiv␈α␈e,␈α
it␈α
is␈α
still␈α	necessary␈α
for␈α
us␈α
to␈α	pro␈α␈v␈α␈e␈α
that␈α
sequences␈α	satisfying
␈β⊂K␈↓ ↓H␈ε"all␈α
these␈α	conditions␈α
exist.␈α⊂In␈α␈tuitiv␈α␈ely,␈α
w␈α␈e␈α
feel␈α
quite␈α
sure␈α
that␈α
there␈α
is␈α
n␈α↓o␈α	problem,
␈β⊂{␈↓ ↓H␈∧⊂{↓Hα↓X
␈β⊃∧␈↓ ↓H␈ε$*A␈α␈t␈α	le␈α␈a␈α↓st,␈α	he␈αλm␈α↓ade␈α	such␈α	a␈α	con␈α␈ten␈α␈tion␈α	when␈α	original␈α␈ly␈α	prepari␈α␈ng␈α
the␈α	ma␈α↓te␈α␈rial␈α	for␈α	this␈α	se␈α␈cti␈α␈o␈α↓n␈α	i␈α␈n
␈β⊃$␈↓ ↓H␈ε$196␈α↓6.
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"157
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα$␈↓ ↓H␈ε"because␈α
w␈α␈e␈α
believ␈α␈e␈α∞that␈α
a␈α∞truly␈α
random␈α
sequence␈α∞exists␈α
and␈α∞satis|es␈α
R6;␈α∞but␈α
a
␈βαO␈↓ ↓H␈ε"pro␈α↓of␈αis␈αreally␈αnecessary␈αto␈αsh␈α↓o␈α␈w␈αthat␈αthe␈αde|nition␈αis␈αconsisten␈α␈t.
␈βαz␈↓ α␈ε"An␈α∞in␈α␈teresting␈α∂meth␈α↓od␈α∞for␈α∂constructing␈α∞sequences␈α∂satisfying␈α∞De|nition␈α∞R5
␈ββ&␈↓ ↓H␈ε"has␈αbeen␈αfound␈αby␈αA.␈↓ ∧%␈ε"Wald,␈αstarting␈↓ ε∃␈ε"with␈αa␈αv␈α␈ery␈αsimple␈α1-distributed␈αsequence.
␈ββa␈↓ ↓H␈ε2Lemma␈α∂T.␈ε/␈α∨Let␈α∂the␈α∂sequence␈α∂of␈α⊂real␈α∂n␈α␈um␈α␈bers␈ε6␈α∂h␈↓ πC␈ε(V␈↓ πk␈ε6i␈ε/␈α∂be␈α⊂de|ned␈α∂in␈α∂terms␈α∂of␈α∂the
␈ββn␈↓ πY␈ε+n
␈β∧␈↓ ↓H␈ε/binary␈αsystem␈αas␈αfollo␈α␈ws:
␈β∧O␈↓ ↓↑␈ε(V␈↓ α
␈ε"=␈α
0,␈↓ β%␈ε(V␈↓ βT␈ε"=␈α
.1,␈↓ ∧v␈ε(V␈↓ ¬%␈ε"=␈α
.01,␈↓ εY␈ε(V␈↓ ππ␈ε"=␈α
.11,␈↓ λ;␈ε(V␈↓ λj␈ε"=␈α
.001,␈↓ 
0␈ε".␈αε.␈αε.
␈β∧[␈↓ ↓s␈ε%0␈↓ β:␈ε%1␈↓ ¬␈ε%2␈↓ εn␈ε%3␈↓ λQ␈ε%4
␈β¬λ␈↓ ¬∃␈ε+r␈↓ ελ␈ε+r␈↓ ε∃␈ε9␈␈ε%1
␈β¬⊂␈↓ ↓↑␈ε(V␈↓ α⊂␈ε"=␈α
.␈↓ αH␈ε(c␈↓ αk␈ε".␈αε.␈αε.␈↓ β≠␈ε(c␈↓ β:␈ε"1␈↓ ∧∀␈ε/if␈↓ ∧6␈ε(n␈ε"␈α	=␈↓ ¬β␈ε"2␈↓ ¬+␈ε"+␈↓ ¬W␈ε(c␈↓ ¬v␈ε"2␈↓ εJ␈ε"+␈↓ εv␈ε6↓␈αε↓␈αε↓␈↓ π(␈ε"+␈↓ πT␈ε(c␈↓ πq␈ε".␈↓ 
p␈ε"(29)
␈β¬≤␈↓ ↓s␈ε+n␈↓ αW␈ε+r␈↓ β*␈ε%1␈↓ ¬f␈ε%1␈↓ πc␈ε+r
␈β¬Z␈↓ ↓H␈ε/Let␈↓ απ␈ε(I␈↓ αn␈ε/den␈α↓ote␈α	the␈α	set␈α	of␈α	all␈α	real␈α
n␈α␈um␈α␈bers␈α	in␈↓ π≡␈ε"[␈↓ π.␈ε"0,␈αε1)␈ε/␈α	wh␈α↓ose␈α	binary␈α	represen␈α␈tation
␈β¬g␈↓ α⊗␈ε+b␈↓ α0␈ε%...␈↓ αK␈ε+b
␈β¬p␈↓ α#␈ε'1␈↓ αX␈ε-r
␈βε¬␈↓ ↓H␈ε/begins␈αwith␈ε"␈α0.␈↓ β$␈ε(b␈↓ βI␈ε".␈αε.␈αε.␈↓ βy␈ε(b␈↓ ∧↔␈ε/;␈αth␈α␈us
␈βε∩␈↓ β4␈ε%1␈↓ ∧	␈ε+r
␈βε1␈↓ ∧y␈ε↓α␈↓ λy␈ε↓↓
␈βεH␈↓ λN␈ε9␈␈↓ λl␈ε+r
␈βεP␈↓ βc␈ε(I␈↓ ∧K␈ε"=␈↓ ¬¬␈ε"(0.␈↓ ¬-␈ε(b␈↓ ¬R␈ε".␈αε.␈αε.␈↓ εα␈ε(b␈↓ ε ␈ε")␈↓ ε;␈ε",␈α(0.␈↓ εy␈ε(b␈↓ π∨␈ε".␈αε.␈αε.␈↓ πO␈ε(b␈↓ πm␈ε")␈↓ λ⊂␈ε"+␈↓ λ<␈ε"2␈↓ 	π␈ε".␈↓ 
p␈ε"(30)
␈βε\␈↓ βq␈ε+b␈↓ ∧␈ε%...␈↓ ∧'␈ε+b␈↓ ¬=␈ε%1␈↓ ε∩␈ε+r␈↓ ε,␈ε%2␈↓ π
␈ε%1␈↓ π←␈ε+r␈↓ πy␈ε%2
␈βεf␈↓ β␈␈ε'1␈↓ ∧4␈ε-r
␈βπ≠␈↓ ↓H␈ε/Then␈αif␈↓ αH␈ε(↔␈↓ α]␈ε"(␈ε(n␈ε")␈ε/␈αden␈α↓otes␈αthe␈αn␈α␈um␈α␈ber␈αof␈↓ ε∧␈ε(V␈↓ ε6␈ε/in␈↓ ε`␈ε(I␈↓ πJ␈ε/for␈ε"␈α0␈ε6␈α
∀␈↓ λL␈ε(k␈↓ λj␈ε"<␈ε(␈α
n␈ε/,␈αw␈α␈e␈αha␈α␈v␈α␈e
␈βπ'␈↓ ε→␈ε+k␈↓ εo␈ε+b␈↓ π	␈ε%...␈↓ π$␈ε+b
␈βπ1␈↓ ε|␈ε'1␈↓ π1␈ε-r
␈βπ]␈↓ εE␈ε9␈␈↓ εb␈ε+r
␈βπe␈↓ ∧}␈ε6j␈↓ ¬λ␈ε(↔␈↓ ¬≥␈ε"(␈ε(n␈ε")/␈ε(n␈ε6␈α∞␈␈↓ ε3␈ε"2␈↓ εp␈ε6j␈α
∀␈ε"␈α
1/␈ε(n␈ε".␈↓ 
p␈ε"(31)
␈βλ*␈↓ λ8␈ε+r
␈βλ0␈↓ ↓H␈ε/Pro␈α↓of.␈ε"␈α~Since␈↓ β!␈ε(↔␈↓ β6␈ε"(␈ε(n␈ε")␈αis␈α
the␈αn␈α␈um␈α␈ber␈α
of␈↓ ε␈ε(k␈↓ ε!␈ε"for␈αwhich␈↓ πA␈ε(k␈↓ π\␈ε"mod␈↓ λ&␈ε"2␈↓ λP␈ε"=␈α(␈↓ 	␈ε(b␈↓ 	/␈ε".␈αε.␈αε.␈↓ 	←␈ε(b␈↓ 	}␈ε")␈↓ 
→␈ε",␈α
w␈α␈e␈αha␈α␈v␈α␈e
␈βλ<␈↓ 	≠␈ε+r␈↓ 	o␈ε%1␈↓ 

␈ε%2
␈βλU␈↓ ¬
␈ε+r␈↓ λ-␈ε+r
␈βλ[␈↓ ↓H␈ε(↔␈↓ ↓]␈ε"(␈ε(n␈ε")␈α
=␈ε(␈α
t␈ε"␈αor␈ε(␈αt␈ε"␈αλ+␈αλ1␈αwhen␈ε6␈αb␈ε(n␈ε"/␈↓ ∧{␈ε"2␈↓ ¬≠␈ε6c␈ε"␈α
=␈ε(␈α
t␈ε".␈α⊂Hence␈ε6␈αj␈↓ ε|␈ε(↔␈↓ π⊃␈ε"(␈ε(n␈ε")␈ε6␈αλ␈␈ε(␈αλn␈ε"/␈↓ λ≠␈ε"2␈↓ λ:␈ε6j␈α
∀␈ε"␈α
1.
␈βλd␈↓ 	@␈∧λd	@≠∂
␈β	⊂␈↓ π≡␈ε+r␈↓ 
`␈ε+r
␈β	⊗␈↓ α␈ε"It␈α∞follo␈α␈ws␈α
from␈α∞(31)␈α
that␈α∞the␈α
sequence␈ε6␈α∞hb␈↓ π␈ε"2␈↓ π+␈ε(V␈↓ πT␈ε6ci␈ε"␈α
is␈α∞an␈α
equidistributed␈↓ 
N␈ε"2␈↓ 
m␈ε"-ary
␈β	#␈↓ πA␈ε+n
␈β	B␈↓ ↓H␈ε"sequence;␈α	hence␈α	by␈αλTheorem␈α	A␈↓ ¬"␈ε",␈ε6␈α	h␈↓ ¬A␈ε(V␈↓ ¬i␈ε6i␈ε"␈α	is␈αλan␈αλequidistributed␈↓ λA␈ε"[␈↓ λQ␈ε"0,␈αε1)␈α	sequence.␈α∞Indeed,
␈β	N␈↓ ¬V␈ε+n
␈β	m␈↓ ↓H␈ε"it␈α
is␈α
prett␈α␈y␈αclear␈α
that␈ε6␈αh␈↓ ∧$␈ε(V␈↓ ∧L␈ε6i␈ε"␈α
is␈αabout␈α
as␈αequidistributed␈α
as␈α
a␈↓ λQ␈ε"[␈↓ λa␈ε"0,␈αε1)␈αsequence␈α
can␈α
be.
␈β	y␈↓ ∧9␈ε+n
␈β
_␈↓ ↓H␈ε"(F␈α⎇or␈α
further␈α
discussion␈α
of␈α
this␈α
and␈α
related␈α
sequences,␈α
see␈α
J.␈α
G.␈↓ 	0␈ε"v␈α⎇an␈α
der␈α
Corput,
␈β
C␈↓ ↓H␈ε/Proc.␈α
Koninklijk␈α␈e␈αNederl.␈αAk␈α⎇ad.␈αWetenschappen␈ε2␈α38␈ε"␈α
(1935),␈α813↑821,␈α1058↑1066;
␈β
n␈↓ ↓H␈ε"J.␈α	H.␈↓ α→␈ε"Halton,␈ε/␈α
Numerische␈α	Math.␈ε2␈α	2␈ε"␈αλ(1960),␈α
84↑90,␈α	196;␈α
L.␈α	H.␈↓ λ\␈ε"Ramsha␈α␈w,␈ε/␈α
J.␈αλNum␈α␈ber
␈β~␈↓ ↓H␈ε/Theory␈ε",␈αto␈αappear.)
␈βE␈↓ α␈ε"No␈α␈w␈α
let␈↓ β∂␈ε6R␈↓ β:␈ε",␈↓ βN␈ε6R␈↓ βy␈ε",␈↓ ∧∞␈ε".␈αε.␈αε.␈↓ ∧A␈ε"be␈α
in|nitely␈α
man␈α␈y␈α
subsequence␈α
rules;␈α
w␈α␈e␈α
seek␈α
a␈α	sequence
␈βQ␈↓ β+␈ε%1␈↓ βj␈ε%2
␈βp␈↓ ↓H␈ε6h␈↓ ↓T␈ε(U␈↓ ↓␈␈ε6i␈ε"␈αfor␈αwhich␈αall␈αthe␈αin|nite␈αsubsequences␈ε6␈αh␈↓ π¬␈ε(U␈↓ π0␈ε6i␈↓ π<␈ε6R␈↓ πt␈ε"are␈αequidistributed.
␈β|␈↓ ↓l␈ε+n␈↓ π≡␈ε+n␈↓ πX␈ε+j
␈β+␈↓ ↓H␈ε2Algorithm␈αW␈ε"␈α(␈ε/Wald␈αsequence␈↓ ¬→␈ε")␈ε2.␈ε"␈α_Giv␈α␈en␈αan␈αin|nite␈αsequence␈αof␈αsubsequence␈αrules
␈βW␈↓ ↓H␈ε6R␈↓ ↓s␈ε",␈↓ αλ␈ε6R␈↓ α3␈ε",␈↓ αH␈ε".␈αε.␈αε.␈↓ α⎇␈ε"that␈αde|ne␈α
subsequences␈αof␈↓ ε0␈ε"[␈↓ ε@␈ε"0,␈αε1)␈αsequences␈αof␈αrational␈α
n␈α␈um␈α␈bers,␈αthis
␈βc␈↓ ↓d␈ε%1␈↓ α$␈ε%2
␈β
α␈↓ ↓H␈ε"procedure␈α⊂de|nes␈α⊂a␈↓ ∧∞␈ε"[␈↓ ∧∨␈ε"0,␈αε1)␈α⊂sequence␈ε6␈α⊂h␈↓ ε∪␈ε(U␈↓ ε?␈ε6i␈ε".␈α≥The␈α⊂computation␈α⊂in␈α␈v␈α␈olv␈α␈es␈α⊂in|nitely
␈β
∞␈↓ ε,␈ε+n
␈β
-␈↓ ↓H␈ε"man␈α␈y␈α
auxiliary␈α
v␈α⎇ariables␈↓ ∧X␈ε(C␈↓ ∧t␈ε"[␈↓ ∧␈␈ε(a␈↓ ¬!␈ε",␈↓ ¬1␈ε".␈αε.␈αε.␈↓ ¬a␈ε",␈↓ ¬q␈ε(a␈↓ ε∩␈ε"],␈α
where␈↓ π≤␈ε(r␈↓ π8␈ε6∃␈ε"␈α1␈α
and␈α
where␈↓ 	7␈ε(a␈↓ 	e␈ε"=␈α0␈α
or␈α
1␈α
for
␈β
9␈↓ ¬∩␈ε%1␈↓ ε∧␈ε+r␈↓ 	J␈ε+j
␈β
X␈↓ ↓H␈ε"1␈ε6␈α
∀␈↓ α∩␈ε(j␈↓ α/␈ε6∀␈↓ α]␈ε(r␈↓ αm␈ε".␈α⊂These␈αv␈α⎇ariables␈αare␈αinitially␈αall␈αzero.
␈β∞␈↓ ↓@␈ε2W1.␈↓ α␈ε"[Initialize␈ε(␈αn␈ε".]␈α→Set␈ε(␈αn␈ε6␈α
␈ ␈ε"␈α
0.
␈β∞?␈↓ ↓@␈ε2W2.␈↓ α␈ε"[Initialize␈↓ β*␈ε(r␈↓ β;␈ε".]␈α→Set␈↓ ∧&␈ε(r␈↓ ∧@␈ε6␈ ␈ε"␈α
1.
␈β∞r␈↓ ↓@␈ε2W3.␈↓ α␈ε"[T␈α⎇est␈↓ αc␈ε6R␈↓ β
␈ε".]␈α⊃If␈α
the␈α	elemen␈α␈t␈↓ ¬∩␈ε(U␈↓ ¬G␈ε"is␈α
to␈α
be␈α
in␈α	the␈α
subsequence␈α
de|ned␈α
by␈↓ 
⊗␈ε6R␈↓ 
@␈ε",␈α
based
␈β∞␈␈↓ α␈␈ε+r␈↓ ¬+␈ε+n␈↓ 
2␈ε+r
␈β∂≡␈↓ α␈ε"on␈αthe␈αv␈α⎇alues␈αof␈↓ ∧⊂␈ε(U␈↓ ∧E␈ε"for␈α0␈ε6␈α
∀␈↓ ¬G␈ε(k␈↓ ¬f␈ε"<␈ε(␈α
n␈ε"␈α␈,␈αset␈↓ εw␈ε(a␈↓ π"␈ε6␈ ␈ε"␈α
1;␈αotherwise␈αset␈↓ 	P␈ε(a␈↓ 	{␈ε6␈ ␈ε"␈α
0.
␈β∂*␈↓ ∧)␈ε+k␈↓ π
␈ε+r␈↓ 	c␈ε+r
␈β∂K␈↓ π(␈ε+r␈↓ π6␈ε9␈␈ε%␈α␈1
␈β∂Q␈↓ ↓@␈ε2W4.␈↓ α␈ε"[␈↓ α⊗␈ε(B␈↓ α2␈ε"[␈↓ α<␈ε(a␈↓ α↑␈ε",␈↓ αn␈ε".␈αε.␈αε.␈↓ β≡␈ε",␈↓ β.␈ε(a␈↓ βO␈ε"]␈↓ βe␈ε"full?]␈α→If␈↓ ∧n␈ε(C␈↓ ¬
␈ε"[␈↓ ¬∃␈ε(a␈↓ ¬7␈ε",␈↓ ¬G␈ε".␈αε.␈αε.␈↓ ¬w␈ε",␈↓ επ␈ε(a␈↓ ε(␈ε"]␈α
<␈α
3␈ε6␈αλ↓␈↓ π⊗␈ε"4␈↓ πb␈ε",␈αgo␈αto␈αW6.
␈β∂]␈↓ αO␈ε%1␈↓ βA␈ε+r␈↓ ¬(␈ε%1␈↓ ε~␈ε+r
␈β⊂∧␈↓ ↓@␈ε2W5.␈↓ α␈ε"[Increase␈↓ β ␈ε(r␈↓ β1␈ε".]␈α→Set␈↓ ∧≤␈ε(r␈↓ ∧6␈ε6␈ ␈↓ ∧d␈ε(r␈↓ ∧⎇␈ε"+␈αλ1␈αand␈αreturn␈αto␈αW3.
␈β⊂8␈↓ ↓@␈ε2W6.␈↓ α␈ε"[Set␈↓ αS␈ε(U␈↓ α␈␈ε".]␈α⊗Increase␈αthe␈αv␈α⎇alue␈αof␈↓ ¬s␈ε(C␈↓ ε⊂␈ε"[␈↓ ε~␈ε(a␈↓ ε<␈ε",␈↓ εL␈ε".␈αε.␈αε.␈↓ ε|␈ε",␈↓ π␈ε(a␈↓ π-␈ε"]␈αby␈α1␈αand␈αlet␈↓ 	
␈ε(k␈↓ 	)␈ε"be␈αits␈αnew␈αv␈α⎇alue.
␈β⊂D␈↓ αl␈ε+n␈↓ ε-␈ε%1␈↓ π∨␈ε+r
␈β⊂c␈↓ α␈ε"Set␈↓ αJ␈ε(U␈↓ α␈␈ε6␈ ␈↓ β-␈ε(V␈↓ βT␈ε",␈αwhere␈↓ ∧R␈ε(V␈↓ ¬∧␈ε"is␈αde|ned␈αin␈αLemma␈αT␈αabo␈α␈v␈α␈e.
␈β⊂o␈↓ αc␈ε+n␈↓ βC␈ε+k␈↓ ∧h␈ε+k
␈β⊃⊗␈↓ ↓@␈ε2W7.␈↓ α␈ε"[Adv␈α⎇ance␈ε(␈αn␈ε".]␈α→Increase␈ε(␈αn␈ε"␈αby␈α1␈αand␈αreturn␈αto␈αW2.
␈β⊃∨␈↓ λ-␈∧⊃∨λ-≠∂
␈β∪(

␈β↓U␈↓ ↓H␈ε"158␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα$␈↓ α␈ε"Strictly␈α⊃speaking,␈α∩this␈α⊂is␈α⊃n␈α↓ot␈α⊃an␈α⊂algorithm,␈α∩since␈α⊃it␈α⊃doesn't␈α⊂terminate;␈α∪it
␈βαO␈↓ ↓H␈ε"w␈α␈ould␈αof␈αcourse␈αbe␈αeasy␈αto␈αm␈α↓odify␈αthe␈αprocedure␈αto␈αstop␈αwhen␈ε(␈αn␈ε"␈αreaches␈αa␈αgiv␈α␈en
␈βαz␈↓ ↓H␈ε"v␈α⎇alue.␈α≠The␈α⊂reader␈α∂will␈α⊂|nd␈α⊂it␈α⊂easier␈α∂to␈α⊂grasp␈α⊂the␈α∂idea␈α⊂of␈α⊂the␈α⊂construction␈α∂by
␈ββ ␈↓ πa␈ε+r␈↓ πn␈ε9␈␈ε%1␈↓ 
+␈ε+r
␈ββ&␈↓ ↓H␈ε"trying␈α
it␈α
out␈α∞man␈α␈ually,␈α∞replacing␈α
the␈α
n␈α␈um␈α␈ber␈α∞3␈ε6␈α	↓␈↓ πO␈ε"4␈↓ λ)␈ε"of␈α
step␈α
W4␈α∞by␈↓ 
→␈ε"2␈↓ 
F␈ε"during
␈ββQ␈↓ ↓H␈ε"this␈αexperimen␈α␈t.
␈ββ|␈↓ α␈ε"Algorithm␈α
W␈α
is␈α
n␈α↓ot␈α
mean␈α␈t␈α∞to␈α
be␈α
a␈α
practical␈α
source␈α
of␈α
random␈α
n␈α␈um␈α␈bers.␈α∪It
␈β∧'␈↓ ↓H␈ε"is␈αin␈α␈tended␈αto␈αserv␈α␈e␈αonly␈αa␈αtheoretical␈αpurpose:
␈β∧]␈↓ ↓H␈ε2Theorem␈α
W.␈ε/␈α∩Let␈ε6␈α
h␈↓ βj␈ε(U␈↓ ∧∃␈ε6i␈ε/␈α
be␈α
the␈α
sequence␈α
of␈α
rational␈α
n␈α␈um␈α␈bers␈α
de|ned␈α
by␈α
Algorithm
␈β∧j␈↓ ∧β␈ε+n
␈β¬λ␈↓ ↓H␈ε/W␈↓ ↓l␈ε/,␈α∂and␈α∂let␈↓ β∧␈ε(k␈↓ β'␈ε/be␈α∂a␈α∞positiv␈α␈e␈α∂in␈α␈teger.␈α_If␈α∞the␈α∂subsequence␈ε6␈α∞h␈↓ λM␈ε(U␈↓ λy␈ε6i␈↓ 	¬␈ε6R␈↓ 	@␈ε/is␈α∂in|nite,␈α∂it␈α∞is
␈β¬∃␈↓ λf␈ε+n␈↓ 	!␈ε+k
␈β¬4␈↓ ↓H␈ε"1␈ε/-distributed␈↓ β∩␈ε/.
␈β¬j␈↓ ↓H␈ε/Pro␈α↓of.␈ε"␈α≤Let␈ε(␈α∞A␈↓ β#␈ε"[␈↓ β-␈ε(a␈↓ βP␈ε",␈↓ β`␈ε".␈αε.␈αε.␈↓ ∧⊂␈ε",␈↓ ∧ ␈ε(a␈↓ ∧@␈ε"]␈α∞den␈α↓ote␈α∞the␈α∞(possibly␈α∞empt␈α␈y)␈α∞subsequence␈α
of␈ε6␈α∞h␈↓ 
%␈ε(U␈↓ 
Q␈ε6i␈ε"␈α
con-
␈β¬v␈↓ β@␈ε%1␈↓ ∧3␈ε+r␈↓ 
>␈ε+n
␈βε∃␈↓ ↓H␈ε"taining␈α∞precisely␈α∞th␈α↓ose␈α∞elemen␈α␈ts␈↓ ¬O␈ε(U␈↓ ελ␈ε"that,␈α∂for␈α∞all␈↓ πQ␈ε(j␈↓ πr␈ε6∀␈↓ λ#␈ε(r␈↓ λ4␈ε",␈α∞belong␈α∞to␈α∞subsequence
␈βε!␈↓ ¬h␈ε+n
␈βε@␈↓ ↓H␈ε6h␈↓ ↓T␈ε(U␈↓ ↓␈␈ε6i␈↓ α␈ε6R␈↓ αB␈ε"if␈↓ αd␈ε(a␈↓ β⊂␈ε"=␈α
1␈αand␈αdo␈αn␈α↓ot␈αbelong␈αto␈αsubsequence␈ε6␈αh␈↓ λ	␈ε(U␈↓ λ5␈ε6i␈↓ λA␈ε6R␈↓ λx␈ε"if␈↓ 	~␈ε(a␈↓ 	F␈ε"=␈α
0.
␈βεL␈↓ ↓l␈ε+n␈↓ α'␈ε+j␈↓ αw␈ε+j␈↓ λ"␈ε+n␈↓ λ]␈ε+j␈↓ 	-␈ε+j
␈βεk␈↓ α␈ε"It␈α∞su}ces␈α∞to␈α∂pro␈α␈v␈α␈e,␈α∞for␈α∂all␈↓ ¬;␈ε(r␈↓ ¬Y␈ε6∃␈ε"␈α∞1␈α∞and␈α∞all␈α∂pairs␈α∞of␈α∞binary␈α∞n␈α␈um␈α␈bers␈↓ 
3␈ε(a␈↓ 
\␈ε".␈αε.␈αε.␈↓ ␈ε(a
␈βεx␈↓ 
F␈ε%1␈↓ ∨␈ε+r
␈βπ⊂␈↓ εa␈ε9␈␈↓ ε␈␈ε+r
␈βπ⊗␈↓ ↓H␈ε"and␈↓ α∀␈ε(b␈↓ α9␈ε".␈αε.␈αε.␈↓ αi␈ε(b␈↓ βπ␈ε",␈α∀that␈↓ βz␈ε"P␈↓ ∧∩␈ε"r␈↓ ∧ ␈ε"(␈↓ ∧,␈ε(U␈↓ ∧l␈ε62␈↓ ¬_␈ε(I␈↓ ¬v␈ε")␈α∃=␈↓ εO␈ε"2␈↓ π∨␈ε"with␈α∩respect␈α∪to␈α∩the␈α∩subsequence
␈βπ#␈↓ α$␈ε%1␈↓ αz␈ε+r␈↓ ∧E␈ε+n␈↓ ¬'␈ε+b␈↓ ¬B␈ε%...␈↓ ¬\␈ε+b␈↓ π?␈ε↓␈␈↓ 	 ␈ε↓↓
␈βπ,␈↓ ¬5␈ε'1␈↓ ¬j␈ε-r
␈βπB␈↓ ↓H␈ε(A␈↓ ↓b␈ε"[␈↓ ↓l␈ε(a␈↓ α∞␈ε",␈↓ α≡␈ε".␈αε.␈αε.␈↓ αN␈ε",␈↓ α↑␈ε(a␈↓ α␈␈ε"],␈α∀whenev␈α␈er␈α∪the␈α∩latter␈α∪is␈α∩in|nite.␈↓ πM␈ε"See␈α∩Eq.␈α∪(30).␈↓ 	@␈ε"F␈α⎇or␈α∪if␈↓ 
0␈ε(r␈↓ 
U␈ε6∃␈↓ ∞␈ε(k␈↓ "␈ε",
␈βπN␈↓ ↓␈␈ε%1␈↓ αq␈ε+r
␈βπm␈↓ ↓H␈ε"the␈α∩in|nite␈α∩sequence␈ε6␈α∪h␈↓ ∧3␈ε(U␈↓ ∧↑␈ε6i␈↓ ∧j␈ε6R␈↓ ¬)␈ε"is␈α∩the␈α∪|nite␈α∩union␈α∩of␈α∪the␈α∩disjoin␈α␈t␈α∩subsequences
␈βπy␈↓ ∧K␈ε+n␈↓ ¬ε␈ε+k
␈βλ_␈↓ ↓H␈ε(A␈↓ ↓b␈ε"[␈↓ ↓l␈ε(a␈↓ α∞␈ε",␈↓ α≡␈ε".␈αε.␈αε.␈↓ αN␈ε",␈↓ α↑␈ε(a␈↓ α␈␈ε"]␈α
for␈↓ βO␈ε(a␈↓ β␈␈ε"=␈α1␈α
and␈↓ ¬∃␈ε(a␈↓ ¬C␈ε"=␈α0␈α
or␈α∞1␈α
for␈α
1␈ε6␈α∀␈↓ πd␈ε(j␈↓ λβ␈ε6∀␈↓ λ3␈ε(r␈↓ λC␈ε",␈↓ λ[␈ε(j␈↓ λz␈ε6≤␈↓ 	)␈ε(k␈↓ 	>␈ε";␈α
and␈α
it␈α
follo␈α␈ws
␈βλ$␈↓ ↓␈␈ε%1␈↓ αq␈ε+r␈↓ βb␈ε+k␈↓ ¬(␈ε+j
␈βλ=␈↓ ∧V␈ε9␈␈↓ ∧t␈ε+r
␈βλC␈↓ ↓H␈ε"that␈↓ α⊗␈ε"P␈↓ α.␈ε"r␈↓ α<␈ε"(␈↓ αH␈ε(U␈↓ α}␈ε62␈↓ β!␈ε(I␈↓ β␈␈ε")␈α=␈↓ ∧D␈ε"2␈↓ ¬∞␈ε"with␈α
respect␈αto␈ε6␈αh␈↓ π∪␈ε(U␈↓ π?␈ε6i␈↓ πK␈ε6R␈↓ πw␈ε".␈α~(See␈α
exercise␈α33.)␈α~This␈αis
␈βλP␈↓ αa␈ε+n␈↓ β0␈ε+b␈↓ βK␈ε%.␈α␈..␈↓ βe␈ε+b␈↓ π,␈ε+n␈↓ πg␈ε+k
␈βλY␈↓ β=␈ε'1␈↓ βs␈ε-r
␈βλn␈↓ ↓H␈ε"en␈α↓ough␈αto␈αsh␈α↓o␈α␈w␈αthat␈αthe␈αsequence␈αis␈α1-distributed,␈αby␈αTheorem␈αA.
␈β	~␈↓ α␈ε"Let␈↓ αN␈ε(B␈↓ αj␈ε"[␈↓ αt␈ε(a␈↓ β⊗␈ε",␈↓ β&␈ε".␈αε.␈αε.␈↓ βV␈ε",␈↓ βf␈ε(a␈↓ ∧π␈ε"]␈αden␈α↓ote␈αthe␈αsubsequence␈αof␈ε6␈α
h␈↓ πQ␈ε(U␈↓ π|␈ε6i␈ε"␈αthat␈αconsists␈αof␈αthe␈αv␈α⎇alues
␈β	&␈↓ βπ␈ε%1␈↓ βy␈ε+r␈↓ πi␈ε+n
␈β	E␈↓ ↓H␈ε"for␈α∞th␈α↓ose␈ε(␈α∂n␈ε"␈α∂in␈α∞which␈↓ ∧ ␈ε(C␈↓ ∧=␈ε"[␈↓ ∧G␈ε(a␈↓ ∧i␈ε",␈↓ ∧y␈ε".␈αε.␈αε.␈↓ ¬)␈ε",␈↓ ¬9␈ε(a␈↓ ¬Z␈ε"]␈α∂is␈α∞increased␈α∂by␈α∂one␈α∂in␈α∞step␈α∂W6␈α∂of␈α∂the␈α∞algo-
␈β	Q␈↓ ∧Z␈ε%1␈↓ ¬L␈ε+r
␈β	j␈↓ 
r␈ε+r␈↓ ␈ε9␈␈ε%␈α␈1
␈β	p␈↓ ↓H␈ε"rithm.␈α⊂By␈αthe␈αalgorithm,␈↓ ∧\␈ε(B␈↓ ∧x␈ε"[␈↓ ¬α␈ε(a␈↓ ¬%␈ε",␈↓ ¬5␈ε".␈αε.␈αε.␈↓ ¬e␈ε",␈↓ ¬u␈ε(a␈↓ ε∃␈ε"]␈αis␈αa␈α|nite␈αsequence␈αwith␈αat␈αm␈α↓ost␈α3␈ε6␈αλ↓␈↓ 
`␈ε"4
␈β	|␈↓ ¬∃␈ε%1␈↓ ελ␈ε+r
␈β
≠␈↓ ↓H␈ε"elemen␈α␈ts.␈α∂All␈αbut␈αa␈α|nite␈αn␈α␈um␈α␈ber␈α
of␈αthe␈αmem␈α␈bers␈αof␈ε(␈αA␈↓ λ→␈ε"[␈↓ λ#␈ε(a␈↓ λE␈ε",␈↓ λU␈ε".␈αε.␈αε.␈↓ 	¬␈ε",␈↓ 	∃␈ε(a␈↓ 	6␈ε"]␈αcome␈αfrom␈α
the
␈β
(␈↓ λ6␈ε%1␈↓ 	(␈ε+r
␈β
F␈↓ ↓H␈ε"subsequences␈↓ β ␈ε(B␈↓ β;␈ε"[␈↓ βF␈ε(a␈↓ βh␈ε",␈↓ βx␈ε".␈αε.␈αε.␈↓ ∧(␈ε",␈↓ ∧8␈ε(a␈↓ ∧Y␈ε",␈↓ ∧i␈ε".␈αε.␈αε.␈↓ ¬→␈ε",␈↓ ¬)␈ε(a␈↓ ¬G␈ε"],␈αwhere␈↓ εO␈ε(a␈↓ ε{␈ε"=␈α
0␈αor␈α1␈αfor␈↓ λI␈ε(r␈↓ λd␈ε"<␈↓ 	∩␈ε(j␈↓ 	/␈ε6∀␈ε(␈α
t␈ε".
␈β
S␈↓ βY␈ε%1␈↓ ∧K␈ε+r␈↓ ¬<␈ε+t␈↓ εb␈ε+j
␈β
r␈↓ α␈ε"No␈α␈w␈α∩assume␈α⊃that␈ε(␈α∩A␈↓ ∧T␈ε"[␈↓ ∧↑␈ε(a␈↓ ¬↓␈ε",␈↓ ¬⊃␈ε".␈αε.␈αε.␈↓ ¬A␈ε",␈↓ ¬Q␈ε(a␈↓ ¬q␈ε"]␈α∩is␈α∩in|nite,␈α∪and␈α∩let␈ε(␈α⊃A␈↓ λa␈ε"[␈↓ λl␈ε(a␈↓ 	∞␈ε",␈↓ 	≡␈ε".␈αε.␈αε.␈↓ 	N␈ε",␈↓ 	↑␈ε(a␈↓ 	␈␈ε"]␈α∪=␈ε6␈α∀h␈↓ 
`␈ε(U␈↓ ⊗␈ε6i␈ε",
␈β
}␈↓ ∧q␈ε%1␈↓ ¬d␈ε+r␈↓ λ␈␈ε%1␈↓ 	q␈ε+r␈↓ 
x␈ε+s
␈βλ␈↓ ε␈ε-n
␈β↔␈↓ λW␈ε+r␈↓ 	0␈ε+q␈↓ 
f␈ε+q␈↓ 
v␈ε%+␈α␈1
␈β≥␈↓ ↓H␈ε"where␈↓ α0␈ε(s␈↓ αZ␈ε"<␈↓ βλ␈ε(s␈↓ β3␈ε"<␈↓ βa␈ε(s␈↓ ∧␈ε6∀␈↓ ∧:␈ε6↓␈αε↓␈αε↓␈↓ ∧j␈ε".␈α⊂If␈↓ ¬)␈ε(N␈↓ ¬X␈ε"is␈αa␈αlarge␈αin␈α␈teger,␈αwith␈↓ λE␈ε"4␈↓ λo␈ε6∀␈↓ 	≡␈ε"4␈↓ 	J␈ε"<␈↓ 	x␈ε(N␈↓ 
%␈ε6∀␈↓ 
T␈ε"4␈↓ "␈ε",
␈β)␈↓ αA␈ε%0␈↓ β→␈ε%1␈↓ βr␈ε%2
␈βH␈↓ ↓H␈ε"it␈α	follo␈α␈ws␈α	that␈α	the␈α
n␈α␈um␈α␈ber␈α	of␈α	v␈α⎇alues␈α	of␈↓ ε→␈ε(k␈↓ ε7␈ε"<␈↓ εe␈ε(N␈↓ π⊃␈ε"for␈α	which␈↓ λ,␈ε(U␈↓ λj␈ε"is␈α	in␈↓ 	2␈ε(I␈↓ 
→␈ε"is␈α	(except
␈βT␈↓ λD␈ε+s␈↓ 	A␈ε+b␈↓ 	\␈ε%.␈α␈..␈↓ 	v␈ε+b
␈β↑␈↓ λR␈ε-k␈↓ 	N␈ε'1␈↓ 
∧␈ε-r
␈βs␈↓ ↓H␈ε"for␈α|nitely␈αman␈α␈y␈αelemen␈α␈ts␈αat␈αthe␈αbeginning␈αof␈αthe␈αsubsequence)
␈β1␈↓ ∧G␈ε(↔␈↓ ∧]␈ε"(␈↓ ∧i␈ε(N␈↓ ¬␈ε")␈α
=␈↓ ¬P␈ε(↔␈↓ ¬e␈ε"(␈↓ ¬q␈ε(N␈↓ ε≡␈ε")␈αλ+␈↓ ε↑␈ε6↓␈αε↓␈αε↓␈↓ π⊂␈ε"+␈↓ π<␈ε(↔␈↓ πR␈ε"(␈↓ π↑␈ε(N␈↓ λ↔␈ε").
␈β>␈↓ ε∂␈ε%1␈↓ π|␈ε+m
␈βo␈↓ ↓H␈ε"Here␈ε(␈α∂m␈ε"␈α∂is␈α∂the␈α∂n␈α␈um␈α␈ber␈α⊂of␈α∂subsequences␈↓ εD␈ε(B␈↓ ε`␈ε"[␈↓ εj␈ε(a␈↓ π␈ε",␈↓ π≤␈ε".␈αε.␈αε.␈↓ πL␈ε",␈↓ π\␈ε(a␈↓ π{␈ε"]␈α∂listed␈α∂abo␈α␈v␈α␈e␈α∂in␈α∂which␈↓ 
x␈ε(U
␈β{␈↓ ε⎇␈ε%1␈↓ πo␈ε+t␈↓ ⊂␈ε+s
␈β
¬␈↓ ≡␈ε-k
␈β
~␈↓ ↓H␈ε"appears␈α
for␈α
some␈↓ βa␈ε(k␈↓ ∧↓␈ε"<␈↓ ∧1␈ε(N␈↓ ∧T␈ε";␈↓ ∧l␈ε(N␈↓ ¬'␈ε"is␈α
the␈α
n␈α␈um␈α␈ber␈α
of␈α
v␈α⎇alues␈α∞of␈↓ λP␈ε(k␈↓ λq␈ε"with␈↓ 	D␈ε(U␈↓ 
ε␈ε"in␈α
the␈α
cor-
␈β
'␈↓ ¬
␈ε+j␈↓ 	]␈ε+s
␈β
0␈↓ 	k␈ε-k
␈β
E␈↓ ↓H␈ε"responding␈αsubsequence;␈αand␈↓ ¬∃␈ε(↔␈↓ ¬*␈ε"(␈↓ ¬6␈ε(N␈↓ ¬d␈ε")␈αis␈αthe␈αn␈α␈um␈α␈ber␈αof␈αsuch␈αv␈α⎇alues␈αthat␈αare␈αalso␈αin
␈β
R␈↓ ¬U␈ε+j
␈β
q␈↓ ↓H␈ε(I␈↓ α&␈ε".␈α⊂Therefore␈αby␈αLemma␈αT␈↓ ¬3␈ε",
␈β
⎇␈↓ ↓V␈ε+b␈↓ ↓q␈ε%...␈↓ α␈ε+b
␈β∞π␈↓ ↓d␈ε'1␈↓ α→␈ε-r
␈β∞#␈↓ β`␈ε9␈␈↓ β⎇␈ε+r␈↓ ε≠␈ε9␈␈↓ ε8␈ε+r␈↓ 	2␈ε9␈␈↓ 	P␈ε+r
␈β∞+␈↓ α?␈ε6j␈↓ αI␈ε(↔␈↓ α←␈ε"(␈↓ αk␈ε(N␈↓ β∞␈ε")␈ε6␈αλ␈␈↓ βN␈ε"2␈↓ ∧␈ε(N␈↓ ∧.␈ε6j␈↓ ∧B␈ε"=␈ε6␈α
j␈↓ ∧z␈ε(↔␈↓ ¬∂␈ε"(␈↓ ¬≠␈ε(N␈↓ ¬I␈ε")␈ε6␈αλ␈␈↓ ε	␈ε"2␈↓ εF␈ε(N␈↓ ε|␈ε"+␈↓ π(␈ε6↓␈αε↓␈αε↓␈↓ πZ␈ε"+␈↓ λε␈ε(↔␈↓ λ≠␈ε"(␈↓ λ'␈ε(N␈↓ λ`␈ε")␈ε6␈αλ␈␈↓ 	 ␈ε"2␈↓ 	↑␈ε(N␈↓ 
↔␈ε6j
␈β∞7␈↓ ¬:␈ε%1␈↓ εe␈ε%1␈↓ λF␈ε+m␈↓ 	|␈ε+m
␈β∞d␈↓ ε≠␈ε9␈␈↓ ε8␈ε+r␈↓ 	F␈ε9␈␈↓ 	d␈ε+r
␈β∞l␈↓ ∧B␈ε6∀␈α
j␈↓ ∧z␈ε(↔␈↓ ¬∂␈ε"(␈↓ ¬≠␈ε(N␈↓ ¬I␈ε")␈ε6␈αλ␈␈↓ ε	␈ε"2␈↓ εF␈ε(N␈↓ εt␈ε6j␈ε"␈αλ+␈↓ π2␈ε6↓␈αε↓␈αε↓␈↓ πd␈ε"+␈ε6␈αλj␈↓ λ~␈ε(↔␈↓ λ/␈ε"(␈↓ λ;␈ε(N␈↓ λt␈ε")␈ε6␈αλ␈␈↓ 	4␈ε"2␈↓ 	r␈ε(N␈↓ 
+␈ε6j
␈β∞x␈↓ ¬:␈ε%1␈↓ εe␈ε%1␈↓ λZ␈ε+m␈↓ 
⊂␈ε+m
␈β∂%␈↓ λ
␈ε+q␈↓ λ~␈ε9␈␈↓ λ7␈ε+r␈↓ λE␈ε%+1␈↓ 	<␈ε+q␈↓ 	L␈ε%+␈α␈1
␈β∂-␈↓ ∧B␈ε6∀␈ε(␈α
m␈ε6␈α
∀␈ε"␈α
1␈αλ+␈αλ2␈αλ+␈αλ4␈αλ+␈↓ π~␈ε6↓␈αε↓␈αε↓␈↓ πL␈ε"+␈↓ πx␈ε"2␈↓ λ|␈ε"<␈↓ 	*␈ε"2␈↓ 	y␈ε".
␈β∂c␈↓ ↓H␈ε"The␈αinequalit␈α␈y␈αon␈ε(␈αm␈ε"␈αfollo␈α␈ws␈αhere␈αfrom␈αthe␈αfact␈αthat,␈αby␈αour␈αch␈α↓oice␈αof␈↓ 
∪␈ε(N␈↓ 
6␈ε",␈↓ 
L␈ε(U␈↓ ∀␈ε"is
␈β∂o␈↓ 
d␈ε+s
␈β∂y␈↓ 
r␈ε-N
␈β⊂∞␈↓ ↓H␈ε"in␈↓ ↓r␈ε(B␈↓ α
␈ε"[␈↓ α_␈ε(a␈↓ α:␈ε",␈↓ αJ␈ε".␈αε.␈αε.␈↓ αz␈ε",␈↓ β
␈ε(a␈↓ β(␈ε"]␈αfor␈αsome␈ε(␈αt␈ε6␈α
∀␈↓ ¬∃␈ε(q␈↓ ¬1␈ε"+␈αλ1.
␈β⊂≠␈↓ α+␈ε%1␈↓ β≥␈ε+t
␈β⊂9␈↓ α␈ε"We␈αha␈α␈v␈α␈e␈αpro␈α␈v␈α␈ed␈αthat
␈β⊂t␈↓ ¬<␈ε9␈␈↓ ¬Y␈ε+r␈↓ ε;␈ε+q␈↓ εK␈ε%+1
␈β⊂u␈↓ λ	␈ε6p
␈β⊂x␈↓ λ'␈∧⊂xλ'α#
␈β⊂|␈↓ βg␈ε6j␈↓ βq␈ε(↔␈↓ ∧ε␈ε"(␈↓ ∧∩␈ε(N␈↓ ∧5␈ε")/␈↓ ∧S␈ε(N␈↓ ∧}␈ε6␈␈↓ ¬*␈ε"2␈↓ ¬g␈ε6j␈α
∀␈↓ ε)␈ε"2␈↓ εx␈ε"/␈↓ π
␈ε(N␈↓ π7␈ε"<␈α
2/␈↓ λ'␈ε(N␈↓ λJ␈ε".
␈β⊃¬␈↓ λ{␈∧⊃¬λ{≠∂
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"159
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα$␈↓ α␈ε"T␈α⎇o␈α
sh␈α↓o␈α␈w␈α
|nally␈α∞that␈α
sequences␈α
satisfying␈α
De|nition␈α
R5␈α∞exist,␈α
w␈α␈e␈α
n␈α↓ote␈α
|rst
␈βαO␈↓ ↓H␈ε"that␈α	if␈ε6␈α
h␈↓ α?␈ε(U␈↓ αk␈ε6i␈ε"␈α	is␈α
a␈↓ β>␈ε"[␈↓ βN␈ε"0,␈αε1)␈α
sequence␈α
of␈α
rational␈α
n␈α␈um␈α␈bers␈α
and␈α	if␈ε6␈α
R␈ε"␈α
is␈α
a␈α
computable␈α	sub-
␈βα\␈↓ αX␈ε+n
␈βαz␈↓ ↓H␈ε"sequence␈αλrule␈α	for␈αλa␈ε(␈α	b␈ε"-ary␈α	sequence,␈α	w␈α␈e␈α	can␈αλmak␈α␈e␈ε6␈α	R␈ε"␈α	in␈α␈to␈αλa␈α	computable␈αλsubsequence
␈ββ ␈↓ α*␈ε90␈↓ ∧z␈ε90␈↓ π
␈ε90
␈ββ&␈↓ ↓H␈ε"rule␈↓ α∞␈ε6R␈↓ α=␈ε"for␈ε6␈αh␈↓ β␈ε(U␈↓ β+␈ε6i␈ε"␈αby␈αletting␈↓ ∧d␈ε(f␈↓ ¬␈ε"(␈↓ ¬_␈ε(x␈↓ ¬;␈ε",␈↓ ¬K␈ε".␈αε.␈αε.␈↓ ¬{␈ε",␈↓ ε␈ε(x␈↓ ε2␈ε")␈αin␈↓ εq␈ε6R␈↓ π ␈ε"equal␈↓ π⎇␈ε(f␈↓ λ!␈ε"(␈ε6b␈ε(b␈↓ λK␈ε(x␈↓ λo␈ε6c␈ε",␈↓ 	
␈ε".␈αε.␈αε.␈↓ 	=␈ε",␈ε6␈αεb␈ε(b␈↓ 	k␈ε(x␈↓ 
⊃␈ε6c␈ε")␈αin␈ε6␈αR␈ε".␈α∂If
␈ββ2␈↓ β→␈ε+n␈↓ ¬,␈ε%1␈↓ ε∨␈ε+n␈↓ λ∂␈ε+n␈↓ λ←␈ε%1␈↓ 	␈␈ε+n
␈ββ6␈↓ ∧z␈ε+n
␈ββK␈↓ ∧X␈ε90
␈ββQ␈↓ ↓H␈ε"the␈↓ απ␈ε"[␈↓ α↔␈ε"0,␈αε1)␈α
sequence␈ε6␈α
h␈↓ ∧¬␈ε(U␈↓ ∧0␈ε6i␈↓ ∧<␈ε6R␈↓ ∧m␈ε"is␈α
equidistributed,␈α
so␈α
is␈α
the␈ε(␈α
b␈ε"-ary␈α
sequence␈ε6␈α
hb␈ε(b␈↓ 
A␈ε(U␈↓ 
l␈ε6ciR␈ε".
␈ββ]␈↓ ∧≥␈ε+n␈↓ 
Z␈ε+n
␈ββ|␈↓ ↓H␈ε"No␈α␈w␈αλthe␈α	set␈α	of␈α	all␈αλcomputable␈α	subsequence␈α	rules␈α	for␈ε(␈αλb␈ε"␈α↓-ary␈α	sequences,␈α	for␈α	all␈αλv␈α⎇alues
␈β∧'␈↓ ↓H␈ε"of␈ε(␈α	b␈ε",␈α
is␈α	coun␈α␈table␈α	(since␈α	only␈α	coun␈α␈tably␈α	man␈α␈y␈α	e{ectiv␈α␈e␈α
algorithms␈α	are␈α	possible),␈α	so
␈β∧R␈↓ ↓H␈ε"they␈αma␈α␈y␈αbe␈αlisted␈αin␈αsome␈αsequence␈↓ ε
␈ε6R␈↓ ε5␈ε",␈↓ εJ␈ε6R␈↓ εv␈ε",␈↓ π␈ε".␈αε.␈αε.␈↓ π;␈ε";␈αtherefore␈αAlgorithm␈αW␈αde|nes
␈β∧←␈↓ ε&␈ε%1␈↓ εf␈ε%2
␈β∧}␈↓ ↓H␈ε"a␈↓ ↓f␈ε"[␈↓ ↓v␈ε"0,␈αε1)␈αsequence␈αthat␈αis␈αrandom␈αin␈αthe␈αsense␈αof␈αDe|nition␈αR5.
␈β¬*␈↓ α␈ε"This␈αλbrings␈α	us␈αλto␈αλa␈αλsomewhat␈α	parado␈α␈xical␈αλsituation.␈α∂As␈αλw␈α␈e␈αλmen␈α␈tioned␈αλearlier,
␈β¬V␈↓ ↓H␈ε"n␈α↓o␈αe{ectiv␈α␈e␈αalgorithm␈αcan␈αde|ne␈αa␈αsequence␈αthat␈αsatis|es␈αDe|nition␈αR4,␈αand␈αfor
␈βε↓␈↓ ↓H␈ε"the␈α
same␈αreason␈αthere␈αis␈α
n␈α↓o␈αe{ectiv␈α␈e␈αalgorithm␈αthat␈α
de|nes␈αa␈αsequence␈α
satisfying
␈βε,␈↓ ↓H␈ε"De|nition␈αR5.␈α⊂A␈αpro␈α↓of␈αof␈αthe␈αexistence␈αof␈αsuch␈αrandom␈αsequences␈αis␈αnecessarily
␈βεW␈↓ ↓H␈ε"n␈α↓onconstructiv␈α␈e;␈αh␈α↓o␈α␈w␈αthen␈αcan␈αAlgorithm␈αW␈αconstruct␈αsuch␈αa␈αsequence?
␈βπ∧␈↓ α␈ε"There␈α	is␈α
n␈α↓o␈α	con␈α␈tradiction␈α	here;␈αw␈α␈e␈α	ha␈α␈v␈α␈e␈α	merely␈α
stum␈α␈bled␈α	on␈α	the␈α
fact␈α	that␈α	the
␈βπ/␈↓ ↓H␈ε"set␈α∞of␈α∞all␈α∂e{ectiv␈α␈e␈α∞algorithms␈α∞cann␈α↓ot␈α∂be␈α∞en␈α␈umerated␈α∞by␈α∂an␈α∞e{ectiv␈α␈e␈α∞algorithm.
␈βπZ␈↓ ↓H␈ε"In␈α⊃other␈α∩w␈α␈ords,␈α∪there␈α∩is␈α∩n␈α↓o␈α⊃e{ectiv␈α␈e␈α∩algorithm␈α∩to␈α⊃select␈α∩the␈↓ 	0␈ε(j␈↓ 	C␈ε"th␈α⊃computable
␈βλ¬␈↓ ↓H␈ε"subsequence␈α⊃rule␈↓ βd␈ε6R␈↓ ∧∂␈ε";␈α∀this␈α⊃happens␈α⊃because␈α⊃there␈α⊃is␈α⊃n␈α↓o␈α⊃e{ectiv␈α␈e␈α⊃algorithm␈α⊃to
␈βλ∩␈↓ ∧␈ε+j
␈βλ1␈↓ ↓H␈ε"determine␈αif␈αa␈αcomputational␈α
meth␈α↓od␈αev␈α␈er␈αterminates.␈α~(We␈αshall␈αreturn␈αto␈αthis
␈βλ\␈↓ ↓H␈ε"topic␈α∞in␈α∂Chapter␈α∞11.)␈α≡Importan␈α␈t␈α∞large␈α∂classes␈α∞of␈α∂algorithms␈ε/␈α∞can␈ε"␈α∂be␈α∞systemati-
␈β	π␈↓ ↓H␈ε"cally␈α∞en␈α␈umerated;␈α∂th␈α␈us,␈α∞for␈α∞example,␈α∂Algorithm␈α∞W␈α∞sh␈α↓o␈α␈ws␈α∞that␈α
it␈α∞is␈α∞possible␈α∞to
␈β	2␈↓ ↓H␈ε"construct,␈αwith␈α
an␈αe{ectiv␈α␈e␈αalgorithm,␈α
a␈αsequence␈α
that␈αsatis|es␈α
De|nition␈αR5␈αif
␈β	]␈↓ ↓H␈ε"w␈α␈e␈αrestrict␈αconsideration␈αto␈αsubsequence␈αrules␈αthat␈αare␈α\primitiv␈α␈e␈αrecursiv␈α␈e."
␈β

␈↓ α␈ε"By␈α
m␈α↓odifying␈α
step␈α
W6␈α
of␈α
Algorithm␈α
W␈↓ ε{␈ε",␈α
so␈α
that␈α
it␈α
sets␈↓ λz␈ε(U␈↓ 	1␈ε6␈ ␈↓ 	`␈ε(V␈↓ 
<␈ε"instead
␈β
⊗␈↓ 	∩␈ε+n␈↓ 	v␈ε+k␈↓ 
ε␈ε%+␈ε+t
␈β
5␈↓ ↓H␈ε"of␈↓ ↓q␈ε(V␈↓ α_␈ε",␈αwhere␈ε(␈αt␈ε"␈αis␈αan␈α␈y␈αn␈α↓onnegativ␈α␈e␈αin␈α␈teger␈αdepending␈αon␈↓ λ3␈ε(a␈↓ λV␈ε",␈↓ λk␈ε".␈αε.␈αε.␈↓ 	≠␈ε",␈↓ 	1␈ε(a␈↓ 	R␈ε",␈αw␈α␈e␈αcan␈αsh␈α↓o␈α␈w
␈β
B␈↓ απ␈ε+k␈↓ λF␈ε%1␈↓ 	D␈ε+r
␈β
`␈↓ ↓H␈ε"that␈αthere␈αare␈ε/␈αuncoun␈α␈tably␈ε"␈αman␈α␈y␈↓ ¬↑␈ε"[␈↓ ¬n␈ε"0,␈αε1)␈αsequences␈αsatisfying␈αDe|nition␈αR5.
␈β≤␈↓ α␈ε"The␈α⊃follo␈α␈wing␈α⊂theorem␈α⊃sh␈α↓o␈α␈ws␈α⊂still␈α⊃an␈α↓other␈α⊃w␈α␈a␈α␈y␈α⊂to␈α⊃pro␈α␈v␈α␈e␈α⊂the␈α⊃existence␈α⊂of
␈βH␈↓ ↓H␈ε"uncoun␈α␈tably␈α⊃man␈α␈y␈α⊃random␈α⊃sequences,␈α∪using␈α⊃a␈α⊃less␈α∩direct␈α⊃argumen␈α␈t␈α⊃based␈α⊃on
␈βs␈↓ ↓H␈ε"measure␈αtheory,␈αev␈α␈en␈αif␈αthe␈αstrong␈αde|nition␈αR6␈αis␈αused:
␈β>␈↓ ↓H␈ε2Theorem␈α⊂M.␈ε/␈α!Let␈α⊃the␈α⊂real␈α⊃n␈α␈um␈α␈ber␈ε(␈α⊂x␈ε/,␈ε"␈α∩0␈ε6␈α⊃∀␈ε(␈α∩x␈ε"␈α⊃<␈α⊃1␈ε/,␈α∩correspond␈α⊃to␈α⊂the␈α⊂binary
␈βi␈↓ ↓H␈ε/sequence␈ε6␈α⊂h␈↓ αl␈ε(X␈↓ β≥␈ε6i␈ε/␈α⊃if␈α⊃the␈α⊃binary␈α⊂represen␈α␈tation␈α⊃of␈ε(␈α⊃x␈ε/␈α⊃is␈ε"␈α⊃(0.␈↓ λ.␈ε(X␈↓ λ[␈ε(X␈↓ 	∞␈ε".␈αε.␈αε.␈↓ 	>␈ε")␈↓ 	Y␈ε/.␈α∨Under␈α⊂this
␈βu␈↓ β
␈ε+n␈↓ λL␈ε%0␈↓ λy␈ε%1␈↓ 	J␈ε%2
␈β
∀␈↓ ↓H␈ε/correspondence,␈αalm␈α↓ost␈αall␈ε(␈αx␈ε/␈αcorrespond␈αto␈αbinary␈αsequences␈↓ 	α␈ε/that␈αare␈αrandom␈αin
␈β
?␈↓ ↓H␈ε/the␈αsense␈αof␈αDe|nition␈αR6.␈ε"␈α⊗(In␈αother␈αw␈α␈ords,␈αthe␈αset␈αof␈αall␈αreal␈ε(␈αx␈ε"␈αthat␈αcorrespond
␈β
k␈↓ ↓H␈ε"to␈αa␈αbinary␈αsequence␈αthat␈αis␈αn␈α↓onrandom␈αby␈αDe|nition␈αR6␈αhas␈αmeasure␈αzero.)
␈β∞>␈↓ ↓H␈ε/Pro␈α↓of.␈ε"␈α≠Let␈ε6␈α∞S␈ε"␈α
be␈α
an␈α∞e{ectiv␈α␈e␈α
algorithm␈α
that␈α∞determines␈α
an␈α∞in|nite␈α
sequence␈α
of
␈β∞j␈↓ ↓H␈ε"distinct␈α
n␈α↓onnegativ␈α␈e␈αin␈α␈tegers␈ε6␈αh␈↓ ¬ ␈ε(s␈↓ ¬C␈ε6i␈ε",␈αwhere␈αthe␈αch␈α↓oice␈αof␈↓ λ≤␈ε(s␈↓ λK␈ε"depends␈α
only␈αon␈ε(␈αn␈ε"␈α
and
␈β∞v␈↓ ¬1␈ε+n␈↓ λ-␈ε+n
␈β∂∃␈↓ ↓H␈ε(X␈↓ α⊃␈ε"for␈α∞0␈ε6␈α∂∀␈↓ β∨␈ε(k␈↓ βB␈ε"<␈ε(␈α∂n␈ε"␈α␈;␈α⊃and␈α∂let␈ε6␈α∂R␈ε"␈α∞be␈α∂a␈α∂computable␈α∂subsequence␈α∂rule.␈α→Then␈α∞an␈α␈y
␈β∂!␈↓ ↓f␈ε+s
␈β∂+␈↓ ↓t␈ε-k
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H␈ε(a␈↓ 
k␈ε",␈↓ ␈ε(a␈↓ "␈ε",
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p␈ε"(32)
␈ββo␈↓ ∧a␈ε%1␈↓ ¬β␈ε%2␈↓ ¬[␈ε+r
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␈β∧/␈↓ πε␈ε%1␈↓ π(␈ε%2␈↓ λ␈ε+r
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=␈α
(0.␈↓ πK␈ε(X␈↓ πx␈ε(X␈↓ λ+␈ε".␈αε.␈αε.␈↓ λ[␈ε")␈↓ 	α␈ε"for␈αwhich␈αthere␈αex-
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∨␈ε",␈↓ 
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␈ε",␈↓ λ∨␈ε(X␈↓ λm␈ε"such␈αthat␈↓ 

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(␈ε+s
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elemen␈α␈t␈αof␈α
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_␈ε"such␈αthat
␈β¬[␈↓ 	π␈ε%0␈↓ 	+␈ε%1␈↓ 	|␈ε%2
␈β¬z␈↓ ↓H␈ε(Y␈↓ αα␈ε"=␈↓ α0␈ε(X␈↓ αt␈ε"for␈α
0␈ε6␈α
∀␈↓ βt␈ε(k␈↓ ∧∩␈ε6∀␈ε(␈α
m␈ε"␈α	also␈α
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is␈α
a␈α	measurable
␈βεπ␈↓ ↓\␈ε+s␈↓ αN␈ε+s␈↓ π∂␈ε%1␈↓ π1␈ε%2␈↓ λ
␈ε+r
␈βε⊂␈↓ ↓j␈ε-k␈↓ α\␈ε-k
␈βε%␈↓ ↓H␈ε"set␈α∞consisting␈α∞of␈α∞the␈α∞|nite␈α∞union␈α∞of␈α∂dy␈α␈adic␈α∞subin␈α␈terv␈α⎇als␈↓ λB␈ε(I␈↓ 	≡␈ε".␈α⊗Since␈α∞there␈α∞are
␈βε2␈↓ λQ␈ε+b␈↓ λk␈ε%...␈↓ 	ε␈ε+b
␈βε;␈↓ λ↑␈ε'1␈↓ 	∪␈ε-t
␈βεQ␈↓ ↓H␈ε"only␈αcoun␈α␈tably␈αman␈α␈y␈αsuch␈αdy␈α␈adic␈αin␈α␈terv␈α⎇als,␈αw␈α␈e␈αsee␈αthat␈↓ λ1␈ε(T␈↓ λK␈ε"(␈↓ λW␈ε(a␈↓ λy␈ε(a␈↓ 	!␈ε".␈αε.␈αε.␈↓ 	Q␈ε(a␈↓ 	r␈ε")␈αis␈αa␈αcoun␈α␈t-
␈βε]␈↓ λj␈ε%1␈↓ 	␈ε%2␈↓ 	d␈ε+r
␈βε|␈↓ ↓H␈ε"able␈αunion␈αof␈αdy␈α␈adic␈αin␈α␈terv␈α⎇als,␈αand␈αit␈αis␈αtherefore␈αmeasurable.␈α⊂F␈α⎇urtherm␈α↓ore,␈αthis
␈βπ'␈↓ ↓H␈ε"argumen␈α␈t␈α∞can␈α∞be␈α
extended␈α∞to␈α∞sh␈α↓o␈α␈w␈α∞that␈α∞the␈α∞measure␈α∞of␈↓ λP␈ε(T␈↓ λj␈ε"(␈↓ λv␈ε(a␈↓ 	≡␈ε".␈αε.␈αε.␈↓ 	N␈ε(a␈↓ 
 ␈ε"0)␈α∞equals
␈βπ3␈↓ 		␈ε%1␈↓ 	a␈ε+r␈↓ 	o␈ε9␈␈ε%␈α␈1
␈βπR␈↓ ↓H␈ε"the␈α⊂measure␈α∂of␈↓ βG␈ε(T␈↓ βa␈ε"(␈↓ βm␈ε(a␈↓ ∧∃␈ε".␈αε.␈αε.␈↓ ∧E␈ε(a␈↓ ¬_␈ε"1),␈α⊂since␈α⊂the␈α⊂latter␈α⊂is␈α⊂a␈α⊂union␈α⊂of␈α⊂dy␈α␈adic␈α∂in␈α␈terv␈α⎇als
␈βπ←␈↓ ∧␈ε%1␈↓ ∧X␈ε+r␈↓ ∧f␈ε9␈␈ε%1
␈βπ⎇␈↓ ↓H␈ε"obtained␈α∂from␈α⊂the␈α⊂former␈α⊂by␈α⊂requiring␈α⊂that␈↓ π_␈ε(Y␈↓ πY␈ε"=␈↓ λ
␈ε(X␈↓ λW␈ε"for␈α⊂0␈ε6␈α⊂∀␈↓ 	j␈ε(k␈↓ 
∞␈ε"<␈ε(␈α⊃m␈ε"␈α∂and
␈βλ
␈↓ π,␈ε+s␈↓ λ+␈ε+s
␈βλ∪␈↓ π:␈ε-k␈↓ λ9␈ε-k
␈βλ)␈↓ ↓H␈ε(Y␈↓ α␈ε6≤␈↓ α9␈ε(X␈↓ α{␈ε".␈α⊂No␈α␈w␈αsince
␈βλ5␈↓ ↓\␈ε+s␈↓ αW␈ε+s
␈βλ?␈↓ ↓j␈ε-m␈↓ αe␈ε-m
␈βλh␈↓ β*␈ε(T␈↓ βD␈ε"(␈↓ βP␈ε(a␈↓ βx␈ε".␈αε.␈αε.␈↓ ∧(␈ε(a␈↓ ∧z␈ε"0)␈ε6␈αλ[␈↓ ¬B␈ε(T␈↓ ¬\␈ε"(␈↓ ¬h␈ε(a␈↓ ε⊂␈ε".␈αε.␈αε.␈↓ ε@␈ε(a␈↓ π∪␈ε"1)␈ε6␈α
∩␈↓ πi␈ε(T␈↓ λα␈ε"(␈↓ λ∞␈ε(a␈↓ λ7␈ε".␈αε.␈αε.␈↓ λg␈ε(a␈↓ 	4␈ε"),
␈βλt␈↓ βc␈ε%1␈↓ ∧;␈ε+r␈↓ ∧I␈ε9␈␈ε%␈α␈1␈↓ ¬{␈ε%1␈↓ εS␈ε+r␈↓ εa␈ε9␈␈ε%1␈↓ λ!␈ε%1␈↓ λz␈ε+r␈↓ 	π␈ε9␈␈ε%1
␈β	'␈↓ ↓H␈ε"the␈α
measure␈α
of␈↓ β@␈ε(T␈↓ βZ␈ε"(␈↓ βf␈ε(a␈↓ ∧λ␈ε(a␈↓ ∧0␈ε".␈αε.␈αε.␈↓ ∧`␈ε(a␈↓ ¬↓␈ε")␈α
is␈α∞at␈α
m␈α↓ost␈α
one-half␈α∞the␈α
measure␈α∞of␈↓ 	K␈ε(T␈↓ 	e␈ε"(␈↓ 	q␈ε(a␈↓ 
→␈ε".␈αε.␈αε.␈↓ 
I␈ε(a␈↓ ⊗␈ε").
␈β	3␈↓ βy␈ε%1␈↓ ∧≠␈ε%2␈↓ ∧s␈ε+r␈↓ 
∧␈ε%1␈↓ 
\␈ε+r␈↓ 
j␈ε9␈␈ε%␈α␈1
␈β	R␈↓ ↓H␈ε"The␈αinequalit␈α␈y␈α(32)␈αfollo␈α␈ws␈αby␈αinduction␈αon␈↓ ε|␈ε(r␈↓ π␈ε".
␈β	⎇␈↓ α␈ε"No␈α␈w␈αthat␈α
(32)␈αhas␈α
been␈αestablished,␈α
the␈αremainder␈α
of␈αthe␈α
pro␈α↓of␈αis␈α
essen␈α␈tially
␈β
(␈↓ ↓H␈ε"to␈α⊂sh␈α↓o␈α␈w␈α⊃that␈α⊃the␈α⊃binary␈α⊃represen␈α␈tations␈α⊃of␈α⊂alm␈α↓ost␈α⊃all␈α⊃real␈α⊃n␈α␈um␈α␈bers␈α⊃are␈α⊂equi-
␈β
T␈↓ ↓H␈ε"distributed.␈α∩The␈α
next␈α
few␈α
paragraphs␈α
constitute␈α
a␈α
rather␈α
long␈α
but␈α
n␈α↓ot␈αdi}cult
␈β
␈␈↓ ↓H␈ε"pro␈α↓of␈α∞of␈α∞this␈α∞fact,␈α∂and␈α∞they␈α∞serv␈α␈e␈α∂to␈α∞illustrate␈α∞t␈α␈ypical␈α∞estimation␈α∞techniques␈α∞in
␈β*␈↓ ↓H␈ε"mathematical␈αanalysis.
␈β:␈↓ ¬l␈ε↓S
␈βU␈↓ α␈ε"F␈α⎇or␈α∞0␈α<␈ε(␈α
∂␈ε"␈α<␈α
1,␈α∞let␈↓ ∧J␈ε(B␈↓ ∧e␈ε"(␈↓ ∧q␈ε(r␈↓ ¬α␈ε",␈ε(␈αε∂␈ε"␈α␈)␈α∞be␈↓ ε⊂␈ε(T␈↓ ε)␈ε"(␈↓ ε5␈ε(a␈↓ ε↑␈ε".␈αε.␈αε.␈↓ π∞␈ε(a␈↓ π.␈ε"),␈α∞where␈α∞the␈α∞union␈α
is␈α∞tak␈α␈en␈α
o␈α␈v␈α␈er
␈βb␈↓ εH␈ε%1␈↓ π!␈ε+r
␈β␈↓ ↓H␈ε"all␈α
binary␈α∞n␈α␈um␈α␈bers␈↓ β␈␈ε(a␈↓ ∧'␈ε".␈αε.␈αε.␈↓ ∧W␈ε(a␈↓ ¬¬␈ε"for␈α∞which␈α
the␈α∞n␈α␈um␈α␈ber␈↓ πk␈ε(↔␈↓ λ␈ε"(␈↓ λ␈ε(r␈↓ λ≥␈ε")␈α
of␈α∞zeros␈α
am␈α↓ong␈↓ 
3␈ε(a␈↓ 
\␈ε".␈αε.␈αε.␈↓ ␈ε(a
␈β
␈↓ ∧∩␈ε%1␈↓ ∧j␈ε+r␈↓ 
F␈ε%1␈↓ ∨␈ε+r
␈β,␈↓ ↓H␈ε"satis|es
␈βR␈↓ ε∂␈ε%1
␈βW␈↓ ¬⊂␈ε6j␈↓ ¬~␈ε(↔␈↓ ¬/␈ε"(␈↓ ¬;␈ε(r␈↓ ¬K␈ε")␈ε6␈αλ␈␈↓ ε#␈ε(r␈↓ ε3␈ε6j␈α
∃␈ε"␈α
1␈αλ+␈ε(␈αλ∂␈↓ πJ␈ε(r␈↓ πZ␈ε".
␈βh␈↓ ε∂␈ε%2
␈βk␈↓ ε∂␈∧kε∂α⊂
␈βr␈↓ π⎇␈ε↓␈␈↓ λ≠␈ε↓↓
␈βv␈↓ πQ␈ε↓P
␈β
λ␈↓ λ␈ε+r
␈β
⊃␈↓ ↓H␈ε"The␈αn␈α␈um␈α␈ber␈αof␈αsuch␈αbinary␈αn␈α␈um␈α␈bers␈αis␈↓ ε5␈ε(C␈↓ εQ␈ε"(␈↓ ε]␈ε(r␈↓ εn␈ε",␈ε(␈αε∂␈ε")␈α
=␈↓ λ5␈ε"summed␈αo␈α␈v␈α␈er␈αthe␈αv␈α⎇alues
␈β
"␈↓ λ␈ε+k
␈β
8␈↓ β1␈ε%1
␈β
<␈↓ ↓H␈ε"of␈↓ ↓p␈ε(k␈↓ α∂␈ε"with␈ε6␈αj␈↓ αj␈ε(k␈↓ β∧␈ε6␈␈↓ βD␈ε(r␈↓ βU␈ε6j␈α
∃␈ε"␈α
1␈α¬+␈ε(␈αε∂␈↓ ∧g␈ε(r␈↓ ∧w␈ε".␈α⊂A␈α
suitable␈αestimate␈αfor␈αthe␈α
tail␈αof␈αthe␈αbin␈α↓omial␈α
dis-
␈β
M␈↓ β1␈ε%2
␈β
P␈↓ β1␈∧
Pβ1α⊂
␈β
g␈↓ ↓H␈ε"tribution␈αcan␈αbe␈αobtained␈αby␈αthe␈αfollo␈α␈wing␈↓ εl␈ε"standard␈αtrick:␈αLet␈ε(␈αx␈ε"␈αbe␈αan␈α␈y␈αpositiv␈α␈e
␈β∞∞␈↓ ¬[␈ε%1
␈β∞∪␈↓ ↓H␈ε"n␈α␈um␈α␈ber␈αless␈αthan␈α1,␈αand␈αlet␈ε(␈αs␈ε"␈α
=␈α
(␈↓ ¬v␈ε"+␈ε(␈αλ∂␈ε")␈↓ ε=␈ε(r␈↓ εM␈ε".␈α⊂Then
␈β∞#␈↓ ¬[␈ε%2
␈β∞'␈↓ ¬[␈∧∞'¬[α⊂
␈β∞F␈↓ 	n␈ε+r
␈β∞I␈↓ αr␈ε↓X␈↓ ∧,␈ε↓X␈↓ ε7␈ε↓X
␈β∞R␈↓ β0␈ε↓∩␈↓ βZ␈ε↓∪␈↓ ∧j␈ε↓∩␈↓ ¬∃␈ε↓∪␈↓ εv␈ε↓∩␈↓ π ␈ε↓∪␈↓ λ`␈ε↓∩␈↓ 	X␈ε↓∪
␈β∞S␈↓ βH␈ε(r␈↓ ¬α␈ε(r␈↓ π∞␈ε(r␈↓ 	A␈ε"1
␈β∞c␈↓ ¬?␈ε+s␈ε9␈␈↓ ¬j␈ε+k␈↓ πJ␈ε+s␈ε9␈␈↓ πv␈ε+k␈↓ λR␈ε+s
␈β∞k␈↓ βz␈ε6∀␈↓ ¬+␈ε(x␈↓ ε¬␈ε6∀␈↓ π6␈ε(x␈↓ λ⊂␈ε"=␈↓ λ>␈ε(x␈↓ λv␈ε"1␈αλ+␈↓ 	|␈ε".
␈β∞␈␈↓ 	@␈∧∞␈	@α∀
␈β∂∧␈↓ βF␈ε(k␈↓ ¬␈ε(k␈↓ π␈ε(k␈↓ 	@␈ε(x
␈β∂≠␈↓ αn␈ε+k␈↓ α}␈ε9∃␈ε+s␈↓ ∧(␈ε+k␈↓ ∧9␈ε9∃␈ε+␈α␈s␈↓ ε3␈ε+k␈↓ εC␈ε9∃␈ε%0
␈β∂I␈↓ πS␈ε+s␈↓ λz␈ε+r
␈β∂O␈↓ ↓H␈ε"By␈α∞elemen␈α␈tary␈α∞calculus,␈α∂the␈α∂minim␈α␈um␈α∞v␈α⎇alue␈α∞of␈↓ π?␈ε(x␈↓ πa␈ε"(1␈α	+␈α
1/␈ε(x␈↓ λn␈ε")␈↓ 	⊗␈ε"occurs␈α∞when␈ε(␈α∂x␈ε"␈α
=
␈β∂z␈↓ ↓H␈ε(r␈↓ ↓X␈ε"/␈ε(s␈ε6␈αλ␈␈ε"␈αλ1␈α
=␈α
(1␈ε6␈αλ␈␈ε"␈αλ2␈ε(∂␈ε")/(1␈αλ+␈αλ2␈ε(∂␈ε"),␈αand␈αthis␈αv␈α⎇alue␈αof␈ε(␈αx␈ε"␈αyields
␈β⊂≡␈↓ ¬\␈ε↓0␈↓ λu␈ε↓1
␈β⊂.␈↓ 	
␈ε+r
␈β⊂E␈↓ β⎇␈ε↓X
␈β⊂N␈↓ ∧d␈ε↓∩␈↓ ¬∞␈ε↓∪
␈β⊂O␈↓ ∧|␈ε(r␈↓ π+␈ε"2
␈β⊂↑␈↓ ¬\␈ε↓@␈↓ λu␈ε↓A
␈β⊂g␈↓ ¬.␈ε6∀␈↓ 	≠␈ε".
␈β⊂|␈↓ ¬x␈∧⊂|¬xααy
␈β⊃↓␈↓ ∧z␈ε(k
␈β⊃ε␈↓ εz␈ε'1␈↓ λ6␈ε'1
␈β⊃	␈↓ π
␈ε%+␈ε+∂␈↓ λG␈ε9␈␈ε+␈α␈∂
␈β⊃∞␈↓ ¬x␈ε"(1␈αλ+␈αλ2␈ε(∂␈↓ εk␈ε")␈↓ π4␈ε"(1␈ε6␈αλ␈␈ε"␈αλ2␈ε(∂␈↓ λ'␈ε")
␈β⊃⊗␈↓ ∧␈ε'1
␈β⊃↔␈↓ εz␈ε'2␈↓ λ6␈ε'2
␈β⊃→␈↓ βO␈ε+k␈↓ β`␈ε9∃␈ε%(␈↓ ∧≤␈ε%+␈ε+␈α␈∂␈ε%␈α↓)␈↓ ∧P␈ε+r␈↓ εz␈∧⊃→εzα∞␈↓ λ6␈∧⊃→λ6α∞
␈β⊃'␈↓ ∧␈ε'2
␈β⊃)␈↓ ∧␈∧⊃)∧α∞
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"161
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα≥␈↓ βC␈ε%1
␈βα!␈↓ ↓H␈ε"No␈α␈w␈αwhen␈ε(␈α∂␈ε"␈α	<␈↓ βc␈ε"w␈α␈e␈αha␈α␈v␈α␈e
␈βα2␈↓ βC␈ε%2
␈βα6␈↓ βC␈∧α6βCα⊂
␈βαr␈↓ π∀␈ε%2␈↓ λ6␈ε%4␈↓ 	W␈ε%6␈↓ ∪␈ε%2
␈βαt␈↓ ↓X␈ε%1␈↓ ∧β␈ε%1␈↓ ε<␈ε%1␈↓ π↑␈ε%1␈↓ λ␈␈ε%1
␈βαx␈↓ ↓H␈ε"(␈↓ ↓n␈ε"+␈ε(␈αβ∂␈ε")␈↓ α6␈ε"ln␈↓ αT␈ε"(1␈αβ+␈αβ2␈ε(∂␈ε")␈αβ+␈αβ(␈↓ ∧→␈ε6␈␈ε(␈α∧∂␈ε"␈α␈)␈↓ ∧a␈ε"ln␈↓ ∧␈␈ε"(1␈ε6␈αβ␈␈ε"␈α∧2␈ε(∂␈ε"␈α␈)␈α
=␈↓ ε[␈ε"(2␈ε(∂␈↓ πλ␈ε")␈↓ π'␈ε"+␈↓ π⎇␈ε"(2␈ε(∂␈↓ λ*␈ε")␈↓ λH␈ε"+␈↓ 	≡␈ε"(2␈ε(∂␈↓ 	K␈ε")␈↓ 	i␈ε"+␈↓ 
⊂␈ε6↓␈αε↓␈αε↓␈↓ 
D␈ε">␈α
2␈↓ ∧␈ε(∂␈↓ "␈ε";
␈ββ	␈↓ ↓X␈ε%2␈↓ ∧β␈ε%2␈↓ ε0␈ε%1␈ε9↓␈ε%2␈↓ πR␈ε%3␈ε9↓␈ε%4␈↓ λs␈ε%5␈ε9↓␈ε%6
␈ββ␈↓ ↓X␈∧β↓Xα⊂␈↓ ∧β␈∧β∧βα⊂␈↓ ε0␈∧βε0α(␈↓ πR␈∧βπRα(␈↓ λs␈∧βλsα(
␈ββJ␈↓ λy␈ε'2
␈ββO␈↓ λ@␈ε9␈␈ε%2␈↓ λm␈ε+∂
␈ββU␈↓ ↓H␈ε"hence␈α∂the␈α∂den␈α↓ominator␈α∂in␈α∂our␈α∂estimate␈α⊂is␈α∂larger␈α∂than␈↓ λ1␈ε(e␈↓ 	π␈ε".␈α→We␈α∂ha␈α␈v␈α␈e␈α∂pro␈α␈v␈α␈ed
␈β∧␈↓ ↓H␈ε"that
␈β∧P␈↓ β∂␈ε↓X
␈β∧Y␈↓ βv␈ε↓∩␈↓ ∧ ␈ε↓∪
␈β∧Z␈↓ ∧∞␈ε(r
␈β∧e␈↓ ¬W␈ε'2
␈β∧j␈↓ ¬␈ε+r␈↓ ¬≡␈ε9␈␈ε%␈α␈2␈↓ ¬J␈ε+∂␈↓ ¬d␈ε+r
␈β∧m␈↓ 	u␈ε%1
␈β∧r␈↓ ∧@␈ε"<␈↓ ∧n␈ε"2␈↓ ¬∞␈ε(e␈↓ ¬r␈ε",␈↓ εJ␈ε"for␈αall␈↓ π4␈ε(r␈↓ πN␈ε">␈α
0␈αand␈α0␈α
<␈ε(␈α
∂␈ε"␈α
<␈↓ 
λ␈ε".␈↓ 
p␈ε"(33)
␈β¬β␈↓ 	u␈ε%2
␈β¬ε␈↓ 	u␈∧¬ε	uα⊂
␈β¬␈↓ ∧␈ε(k
␈β¬!␈↓ β≡␈ε'1
␈β¬$␈↓ αb␈ε+k␈↓ αr␈ε9∃␈ε%(␈↓ β.␈ε%+␈ε+∂␈ε%)␈↓ βc␈ε+r
␈β¬2␈↓ β≡␈ε'2
␈β¬4␈↓ β≡␈∧¬4β≡α∞
␈β¬`␈↓ ↓H␈ε"But␈↓ α∞␈ε(C␈↓ α*␈ε"(␈↓ α6␈ε(r␈↓ αF␈ε",␈ε(␈αε∂␈ε")␈αis␈αt␈α␈wice␈αthis␈αle$-hand␈αquan␈α␈tit␈α␈y,␈αhence␈αby␈α(32)␈αand␈α(33)
␈βε1␈↓ λ{␈ε'2
␈βε6␈↓ εY␈ε9␈␈↓ εw␈ε+r␈↓ λA␈ε9␈␈ε%2␈↓ λn␈ε+∂␈↓ 	λ␈ε+r
␈βε>␈↓ βT␈ε(B␈↓ βp␈ε"(␈↓ β|␈ε(r␈↓ ∧␈ε",␈ε(␈αε∂␈ε")␈↓ ∧C␈ε"has␈αmeasure␈↓ ε→␈ε6∀␈↓ εG␈ε"2␈↓ π¬␈ε(C␈↓ π!␈ε"(␈↓ π-␈ε(r␈↓ π=␈ε",␈ε(␈αε∂␈ε")␈α
<␈α
2␈↓ λ2␈ε(e␈↓ 	⊗␈ε".
␈βπ∃␈↓ α␈ε"The␈αnext␈αstep␈αis␈αto␈αde|ne
␈βπl␈↓ β-␈ε(B␈↓ βI␈ε"*(␈↓ βg␈ε(r␈↓ βw␈ε",␈ε(␈αε∂␈ε")␈α
=␈↓ ∧Z␈ε(B␈↓ ∧v␈ε"(␈↓ ¬α␈ε(r␈↓ ¬∩␈ε",␈ε(␈αε∂␈ε")␈ε6␈αλ[␈↓ ¬g␈ε(B␈↓ εβ␈ε"(␈↓ ε∂␈ε(r␈↓ ε'␈ε"+␈αλ1,␈ε(␈αε∂␈ε")␈ε6␈αλ[␈↓ π:␈ε(B␈↓ πU␈ε"(␈↓ πa␈ε(r␈↓ πz␈ε"+␈αλ2,␈ε(␈αε∂␈ε")␈ε6␈αλ[␈↓ 	
␈ε6↓␈αε↓␈αε↓␈↓ 	=␈ε".
␈βλ-␈↓ ¬z␈ε↓P
␈βλ=␈↓ π>␈ε'2
␈βλA␈↓ π¬␈ε9␈␈ε%2␈↓ π2␈ε+∂␈↓ πK␈ε+r
␈βλH␈↓ ↓H␈ε"The␈α
measure␈α
of␈↓ βK␈ε(B␈↓ βg␈ε"*(␈↓ ∧¬␈ε(r␈↓ ∧∃␈ε",␈ε(␈αε∂␈ε")␈α
is␈α
at␈α
m␈α↓ost␈↓ εb␈ε(k␈↓ εv␈ε(e␈↓ πY␈ε",␈α∞and␈α
this␈α
is␈α
the␈α
remainder␈α
of
␈βλY␈↓ ε ␈ε+k␈↓ ε0␈ε9∃␈↓ εN␈ε+r
␈βλs␈↓ ↓H␈ε"a␈αcon␈α␈v␈α␈ergen␈α␈t␈αseries␈αso
␈β	+␈↓ ¬⊃␈ε↓␈␈↓ πJ␈ε↓↓
␈β	J␈↓ ∧T␈ε"lim␈↓ ¬∨␈ε"measure␈αof␈↓ εU␈ε(B␈↓ εq␈ε"*(␈↓ π∂␈ε(r␈↓ π∨␈ε",␈ε(␈αε∂␈ε")␈↓ πb␈ε"=␈α
0.␈↓ 
p␈ε"(34)
␈β	j␈↓ ∧H␈ε+r␈↓ ∧V␈ε9!1
␈β
1␈↓ ↓H␈ε"No␈α␈w␈α⊂if␈ε(␈α⊂x␈ε"␈α⊂is␈α⊂a␈α⊂real␈α⊃n␈α␈um␈α␈ber␈α⊂wh␈α↓ose␈α⊂binary␈α⊂expansion␈α⊂(0.␈↓ λ<␈ε(X␈↓ λj␈ε(X␈↓ 	≥␈ε".␈αε.␈αε.␈↓ 	M␈ε")␈↓ 	x␈ε"leads␈α⊂to␈α⊂an
␈β
=␈↓ λZ␈ε%0␈↓ 	λ␈ε%1␈↓ 	Y␈ε%2
␈β
\␈↓ ↓H␈ε"in|nite␈α	sequence␈ε6␈α	h␈↓ β\␈ε(X␈↓ ∧_␈ε6iR␈ε"␈α	that␈α
is␈α	n␈α↓ot␈α	1-distributed,␈α
and␈α	if␈↓ λ4␈ε(↔␈↓ λI␈ε"(␈↓ λU␈ε(r␈↓ λf␈ε")␈α	den␈α↓otes␈α	the␈α	n␈α␈um␈α␈ber
␈β
h␈↓ βz␈ε+s
␈β
r␈↓ ∧λ␈ε-n
␈βπ␈↓ ↓H␈ε"of␈αzeros␈αin␈αthe␈α|rst␈↓ β}␈ε(r␈↓ ∧~␈ε"elemen␈α␈ts␈αof␈αthe␈αlatter␈αsequence,␈αthen
␈βT␈↓ εJ␈ε"1
␈βl␈↓ ¬(␈ε6j␈↓ ¬2␈ε(↔␈↓ ¬G␈ε"(␈↓ ¬S␈ε(r␈↓ ¬c␈ε")/␈↓ ε↓␈ε(r␈↓ ε~␈ε6␈␈↓ ε`␈ε6j␈α
∃␈ε"␈α
2␈ε(∂␈ε"␈α␈,
␈β↓␈↓ εJ␈∧↓εJα∩
␈βε␈↓ εJ␈ε"2
␈βP␈↓ ↓H␈ε"for␈α
some␈ε(␈α
∂␈ε"␈α>␈α0␈α
and␈α
in|nitely␈α
man␈α␈y␈↓ ελ␈ε(r␈↓ ε→␈ε".␈α∪This␈α
means␈ε(␈α∞x␈ε"␈α
is␈α
in␈↓ λj␈ε(B␈↓ 	ε␈ε"*(␈↓ 	$␈ε(r␈↓ 	4␈ε",␈ε(␈αε∂␈ε")␈α
for␈α
all␈↓ 
X␈ε(r␈↓ 
i␈ε".␈α∪So
␈β{␈↓ ↓H␈ε"|nally␈αw␈α␈e␈α|nd␈αthat
␈β
¬␈↓ ε␈ε↓[␈↓ εP␈ε↓\
␈β
'␈↓ ∧K␈ε(N␈↓ ∧n␈ε"(␈ε6R␈ε",␈ε6␈αεS␈ε")␈α
=␈↓ πλ␈ε(B␈↓ π#␈ε"*(␈↓ πA␈ε(r␈↓ πR␈ε",␈αε1/␈ε(t␈ε");
␈β
W␈↓ εβ␈ε+t␈ε9∃␈ε%2␈↓ εG␈ε+r␈↓ εU␈ε9∃␈ε%␈α␈1
␈β
v␈↓ β+␈ε↓T
␈β∞⊃␈↓ ↓H␈ε"and,␈α⊃by␈α⊂(34),␈↓ ∧	␈ε(B␈↓ ∧%␈ε"*(␈↓ ∧C␈ε(r␈↓ ∧T␈ε",␈αε1/␈ε(t␈ε")␈α⊂has␈α⊂measure␈α⊂zero␈α⊂for␈α⊂all␈ε(␈α⊂t␈ε".␈α≥Hence␈↓ 	i␈ε(N␈↓ 
␈ε"(␈ε6R␈ε",␈ε6␈αεS␈ε"␈α␈)␈α⊂has
␈β∞#␈↓ βI␈ε+r␈↓ βW␈ε9∃␈ε%␈α␈1
␈β∞<␈↓ ↓H␈ε"measure␈αzero.
␈β∞E␈↓ βE␈∧∞EβE≠∂
␈β∞}␈↓ α␈ε"F␈α⎇rom␈α⊃the␈α⊃existence␈α⊃of␈ε/␈α⊃binary␈ε"␈α∩sequences␈α⊃satisfying␈α⊃De|nition␈α⊃R6,␈α∩w␈α␈e␈α⊃can
␈β∂)␈↓ ↓H␈ε"sh␈α↓o␈α␈w␈α
the␈αexistence␈αof␈↓ ∧∨␈ε"[␈↓ ∧/␈ε"0,␈αε1)␈αsequences␈αthat␈αare␈αrandom␈α
in␈αthis␈αsense.␈α⊂F␈α⎇or␈α
details,
␈β∂U␈↓ ↓H␈ε"see␈αexercise␈α36.␈α⊂The␈αconsistency␈αof␈αDe|nition␈αR6␈αis␈αthereby␈αestablished.
␈β⊂⊗␈↓ ↓H␈ε2E.␈α
Random␈α∞|nite␈α
sequences.␈ε"␈α≤An␈α
argumen␈α␈t␈↓ πα␈ε"w␈α␈as␈α
giv␈α␈en␈α∞abo␈α␈v␈α␈e␈α
to␈α∞indicate␈α
that␈α
it
␈β⊂B␈↓ ↓H␈ε"is␈αimpossible␈αto␈αde|ne␈αthe␈αconcept␈αof␈αrandomness␈αfor␈α|nite␈αsequences;␈αan␈α␈y␈αgiv␈α␈en
␈β⊂m␈↓ ↓H␈ε"|nite␈αsequence␈α
is␈αas␈α
lik␈α␈ely␈α
as␈αan␈α␈y␈α
other.␈α⊃Still,␈α
nearly␈α
ev␈α␈ery␈α␈one␈αw␈α␈ould␈α
agree␈αthat
␈β⊃_␈↓ ↓H␈ε"the␈αsequence␈α
011101001␈αis␈α
\m␈α↓ore␈αrandom"␈α
than␈α101010101,␈α
and␈αev␈α␈en␈α
the␈αlatter
␈β∪(

␈β↓U␈↓ ↓H␈ε"162␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα$␈↓ ↓H␈ε"sequence␈αis␈α\m␈α↓ore␈αrandom"␈αthan␈α000000000.␈α⊂Alth␈α↓ough␈αit␈αis␈αtrue␈αthat␈αtruly␈αran-
␈βαO␈↓ ↓H␈ε"dom␈α
sequences␈α∞will␈α∞exhibit␈↓ ∧␈␈ε"locally␈α∞n␈α↓onrandom␈α∞beha␈α␈vior,␈α∞w␈α␈e␈α∞w␈α␈ould␈α∞expect␈α
such
␈βαz␈↓ ↓H␈ε"beha␈α␈vior␈αonly␈αin␈αa␈αlong␈α|nite␈αsequence,␈αn␈α↓ot␈αin␈αa␈αsh␈α↓ort␈αone.
␈ββ&␈↓ α␈ε"Sev␈α␈eral␈α⊃w␈α␈a␈α␈ys␈α⊃for␈α⊃de|ning␈α∩the␈α⊃randomness␈α⊃of␈α⊃a␈α⊃|nite␈α⊃sequence␈α⊃ha␈α␈v␈α␈e␈α⊃been
␈ββQ␈↓ ↓H␈ε"proposed,␈α
and␈αonly␈α
a␈αfew␈α
of␈αthe␈α
ideas␈α
will␈αbe␈α
sk␈α␈etched␈αhere.␈α∂Let␈αus␈α
consider␈α
only
␈ββ|␈↓ ↓H␈ε"the␈αcase␈αof␈ε(␈αb␈ε"-ary␈αsequences.
␈β∧'␈↓ α␈ε"Giv␈α␈en␈αa␈ε(␈αb␈ε"-ary␈αsequence␈↓ ¬↓␈ε(X␈↓ ¬/␈ε",␈↓ ¬E␈ε(X␈↓ ¬r␈ε",␈↓ ελ␈ε".␈αε.␈αε.␈↓ ε8␈ε",␈↓ εN␈ε(X␈↓ ππ␈ε",␈αw␈α␈e␈αcan␈αsa␈α␈y␈↓ λR␈ε"that
␈β∧4␈↓ ¬∨␈ε%1␈↓ ¬c␈ε%2␈↓ εl␈ε+N
␈β∧d␈↓ βv␈ε↓␈␈↓ ∧K␈ε↓↓
␈β∧|␈↓ λZ␈ε6p
␈β∧␈␈↓ λx␈∧∧␈λxα#
␈β¬β␈↓ βO␈ε"P␈↓ βg␈ε"r␈↓ ∧∧␈ε(S␈↓ ∧≥␈ε"(␈ε(n␈ε")␈↓ ∧c␈ε6→␈ε(␈α
p␈ε",␈↓ ¬|␈ε"if␈↓ ε≡␈ε6j␈↓ ε(␈ε(↔␈↓ ε=␈ε"(␈↓ εI␈ε(N␈↓ εl␈ε")/␈↓ π
␈ε(N␈↓ π5␈ε6␈␈ε(␈αλp␈ε6j␈α
∀␈ε"␈α
1/␈↓ λx␈ε(N␈↓ 	≠␈ε",
␈β¬Z␈↓ ↓H␈ε"where␈↓ α3␈ε(↔␈↓ αH␈ε"(␈ε(n␈ε")␈α∂is␈α⊂the␈α∂quan␈α␈tit␈α␈y␈α⊂appearing␈α∂in␈α⊂De|nition␈α∂A␈α⊂at␈α∂the␈α⊂beginning␈α∂of␈α∂this
␈βε¬␈↓ ↓H␈ε"section.␈α⊂The␈αabo␈α␈v␈α␈e␈αsequence␈αcan␈αbe␈αcalled␈α\␈↓ ε{␈ε(k␈↓ π∂␈ε"-distributed"␈αif
␈βεT␈↓ 	≠␈ε+k
␈βε\␈↓ βH␈ε"P␈↓ β`␈ε"r␈↓ βo␈ε"(␈↓ β{␈ε(X␈↓ ∧+␈ε(X␈↓ ¬∂␈ε".␈αε.␈αε.␈↓ ¬?␈ε(X␈↓ εT␈ε"=␈↓ πα␈ε(x␈↓ π%␈ε(x␈↓ πN␈ε".␈αε.␈αε.␈↓ π}␈ε(x␈↓ λ#␈ε")␈ε6␈α
→␈ε"␈α
1/␈↓ 	␈ε(b
␈βεh␈↓ ∧→␈ε+n␈↓ ∧I␈ε+n␈ε%+␈α␈1␈↓ ¬]␈ε+n␈ε%␈α␈+␈↓ ε
␈ε+k␈↓ ε≥␈ε9␈␈ε%1␈↓ π⊗␈ε%1␈↓ π9␈ε%2␈↓ λ∩␈ε+k
␈βπ∀␈↓ ¬q␈ε↓␈
␈βπ2␈↓ ↓H␈ε"for␈α∞all␈ε(␈α∂b␈ε"-ary␈α∂n␈α␈um␈α␈bers␈↓ ∧(␈ε(x␈↓ ∧K␈ε(x␈↓ ∧t␈ε".␈αε.␈αε.␈↓ ¬$␈ε(x␈↓ ¬I␈ε".␈↓ ¬␈␈ε"Cf.␈α∞De|nition␈α∂D␈↓ λ∧␈ε";␈α⊂unfortunately␈α∂a␈α∞sequence
␈βπ?␈↓ ∧<␈ε%1␈↓ ∧←␈ε%2␈↓ ¬8␈ε+k␈↓ ≤␈ε↓↓
␈βπ↑␈↓ ↓H␈ε"ma␈α␈y␈αbe␈↓ αF␈ε(k␈↓ αZ␈ε"-distributed␈αby␈αthis␈αnew␈αde|nition␈αwhen␈αit␈αis␈αn␈α↓ot␈α(␈↓ λt␈ε(k␈↓ 	⊂␈ε6␈␈ε"␈αλ1)-distributed.
␈βλ	␈↓ α␈ε"A␈α∞de|nition␈α
of␈α∞randomness␈α∞ma␈α␈y␈α
n␈α↓o␈α␈w␈α∞be␈α∞giv␈α␈en␈α
analogous␈α∞to␈α∞De|nition␈α
R1,
␈βλ4␈↓ ↓H␈ε"as␈αfollo␈α␈ws:
␈βλv␈↓ ↓H␈ε2De|nition␈α
Q1.␈ε/␈α≠A␈ε(␈αb␈ε/␈α↓-ary␈α
sequence␈α
of␈α
length␈↓ εv␈ε(N␈↓ π%␈ε/is␈α
\random"␈α
if␈α
it␈α
is␈↓ 	`␈ε(k␈↓ 	t␈ε/-distributed
␈β	!␈↓ ↓H␈ε/(in␈αthe␈αabo␈α␈v␈α␈e␈αsense)␈αfor␈αall␈αpositiv␈α␈e␈αin␈α␈tegers␈↓ ε|␈ε(k␈↓ π~␈ε6∀␈↓ πH␈ε"log␈↓ λ	␈ε(N␈↓ λ,␈ε/.
␈β	1␈↓ πv␈ε+b
␈β	b␈↓ α␈ε"According␈α
to␈α
this␈α∞de|nition,␈α
for␈α
example,␈α∞there␈α
are␈α
170␈α
n␈α↓onrandom␈α
binary
␈β

␈↓ ↓H␈ε"sequences␈αof␈αlength␈α11:
␈β
[␈↓ αB␈ε"00000001111␈↓ ∧P␈ε"10000000111␈↓ ε↑␈ε"11000000011␈↓ λl␈ε"11100000001
␈βε␈↓ αB␈ε"00000001110␈↓ ∧P␈ε"10000000110␈↓ ε↑␈ε"11000000010␈↓ λl␈ε"11100000000
␈β1␈↓ αB␈ε"00000001101␈↓ ∧P␈ε"10000000101␈↓ ε↑␈ε"11000000001␈↓ λl␈ε"10100000001
␈β\␈↓ αB␈ε"00000001011␈↓ ∧P␈ε"10000000011␈↓ ε↑␈ε"01000000011␈↓ λl␈ε"01100000001
␈βλ␈↓ αB␈ε"00000000111
␈βP␈↓ ↓H␈ε"plus␈α
01010101010␈α
and␈α
all␈α
sequences␈α
with␈α
nine␈α∞or␈α
m␈α↓ore␈α
zeros,␈α
plus␈α
all␈α
sequences
␈β{␈↓ ↓H␈ε"obtained␈αfrom␈αthe␈αpreceding␈αsequences␈αby␈αin␈α␈terchanging␈αones␈αand␈αzeros.
␈β
&␈↓ α␈ε"Similarly,␈α⊃w␈α␈e␈α⊂can␈α⊂form␈α␈ulate␈α⊂a␈α⊂de|nition␈α⊂for␈α⊂|nite␈α⊂sequences␈α⊂analogous␈α⊂to
␈β
Q␈↓ ↓H␈ε"De|nition␈α∂R6.␈α≠Let␈ε2␈α∂A␈ε"␈α⊂be␈α⊂a␈α∂set␈α⊂of␈α∂algorithms,␈α⊃each␈α∂of␈α⊂which␈α∂is␈α⊂a␈α∂com␈α␈bination
␈β
⎇␈↓ ↓H␈ε"selection␈αand␈αch␈α↓oice␈αprocedure␈αthat␈αgiv␈α␈es␈αa␈αsubsequence␈ε6␈αh␈↓ λL␈ε(X␈↓ 	λ␈ε6iR␈ε",␈αas␈αin␈αthe␈αpro␈α↓of
␈β∞	␈↓ λj␈ε+s
␈β∞∪␈↓ λx␈ε-n
␈β∞(␈↓ ↓H␈ε"of␈αTheorem␈αM.
␈β∞i␈↓ ↓H␈ε2De|nition␈α	Q2.␈ε/␈α⊃The␈ε(␈α
b␈ε/-ary␈α
sequence␈↓ ¬i␈ε(X␈↓ ε⊗␈ε/,␈↓ ε*␈ε(X␈↓ εW␈ε/,␈↓ εk␈ε".␈αε.␈αε.␈↓ π≠␈ε/,␈↓ π/␈ε(X␈↓ πs␈ε/is␈ε"␈α
(␈ε(n␈ε"␈α␈,␈ε(␈αε∂␈ε")␈ε/-random␈α
with␈α	respect
␈β∞v␈↓ επ␈ε%1␈↓ εH␈ε%2␈↓ πM␈ε+N
␈β∂∀␈↓ ↓H␈ε/to␈αa␈αset␈αof␈αalgorithms␈ε2␈αA␈↓ ∧=␈ε/,␈αif␈αfor␈αev␈α␈ery␈αsubsequence␈↓ πS␈ε(X␈↓ λ
␈ε/,␈↓ λ∨␈ε(X␈↓ λV␈ε/,␈↓ λl␈ε".␈αε.␈αε.␈↓ 	≤␈ε/,␈↓ 	1␈ε(X␈↓ 	|␈ε/determined
␈β∂!␈↓ πq␈ε+t␈↓ λ=␈ε+t␈↓ 	O␈ε+t
␈β∂*␈↓ π⎇␈ε'1␈↓ λI␈ε'2␈↓ 	[␈ε-m
␈β∂@␈↓ ↓H␈ε/by␈αan␈αalgorithm␈αof␈ε2␈αA␈ε/␈αw␈α␈e␈αha␈α␈v␈α␈e␈αeither␈ε(␈αm␈ε"␈α	<␈ε(␈α
n␈ε/␈αor
␈β∂v␈↓ β7␈ε↓␈␈↓ ε;␈ε↓␈
␈β⊂␈↓ β7␈ε↓␈␈↓ ε;␈ε↓␈
␈β⊂∩␈↓ βN␈ε"1␈↓ ε%␈ε"1
␈β⊂!␈↓ β7␈ε↓␈␈↓ ε;␈ε↓␈
␈β⊂*␈↓ βk␈ε(↔␈↓ ∧∞␈ε"(␈↓ ∧~␈ε(X␈↓ ∧Q␈ε",␈↓ ∧a␈ε".␈αε.␈αε.␈↓ ¬⊃␈ε",␈↓ ¬!␈ε(X␈↓ ¬a␈ε")␈ε6␈αλ␈␈↓ εQ␈ε6∀␈ε(␈α
∂␈↓ πU␈ε/for␈↓ λ
␈ε"0␈ε6␈α
∀␈ε(␈α
a␈ε"␈α
<␈ε(␈α
b␈ε"␈α↓.
␈β⊂6␈↓ β}␈ε+a␈↓ ∧8␈ε+t␈↓ ¬?␈ε+t
␈β⊂7␈↓ β7␈ε↓␈␈↓ ε;␈ε↓␈
␈β⊂>␈↓ βG␈∧⊂>βGα ␈↓ ε%␈∧⊂>ε%α∩
␈β⊂@␈↓ ∧D␈ε'1␈↓ ¬J␈ε-m
␈β⊂D␈↓ βG␈ε(m␈↓ ε%␈ε(b
␈β⊃⊗␈↓ ↓H␈ε/Here␈↓ α≤␈ε(↔␈↓ α?␈ε"(␈↓ αK␈ε(x␈↓ αn␈ε",␈↓ α}␈ε".␈αε.␈αε.␈↓ β.␈ε",␈↓ β>␈ε(x␈↓ βl␈ε")␈ε/␈αis␈αthe␈αn␈α␈um␈α␈ber␈αof␈ε(␈αa␈ε/'s␈αin␈αthe␈αsequence␈↓ λE␈ε(x␈↓ λi␈ε/,␈↓ λ␈␈ε".␈αε.␈αε.␈↓ 	/␈ε/,␈↓ 	E␈ε(x␈↓ 	s␈ε/.
␈β⊃#␈↓ α.␈ε+a␈↓ α←␈ε%1␈↓ βR␈ε+m␈↓ λY␈ε%1␈↓ 	Y␈ε+m
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"163
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα"␈↓ α␈ε"(In␈αother␈αw␈α␈ords,␈αev␈α␈ery␈α
su}cien␈α␈tly␈αlong␈αsubsequence␈αdetermined␈αby␈αan␈αalgo-
␈βαM␈↓ ↓H␈ε"rithm␈αof␈ε2␈α
A␈ε"␈αm␈α␈ust␈α
be␈αappro␈α␈ximately␈α
equidistributed.)␈α∩The␈αbasic␈α
idea␈αin␈α
this␈αcase
␈βαy␈↓ ↓H␈ε"is␈αto␈α
let␈ε2␈αA␈ε"␈α
be␈αa␈α
set␈α
of␈α\simple"␈α
algorithms;␈α
the␈αn␈α␈um␈α␈ber␈α
(and␈αthe␈α
complexit␈α␈y)␈αof
␈ββ$␈↓ ↓H␈ε"the␈αalgorithms␈αin␈ε2␈αA␈ε"␈αcan␈αgro␈α␈w␈αas␈↓ ¬M␈ε(N␈↓ ¬|␈ε"gro␈α␈ws.
␈ββO␈↓ α␈ε"As␈αan␈αexample␈αof␈αDe|nition␈αQ2,␈αlet␈αus␈αconsider␈αbinary␈αsequences,␈αand␈αlet␈ε2␈αA
␈ββz␈↓ ↓H␈ε"be␈αjust␈αthe␈αfollo␈α␈wing␈αfour␈αalgorithms:
␈β∧9␈↓ ↓b␈ε"a)␈↓ α␈ε"T␈α⎇ak␈α␈e␈αthe␈αwh␈α↓ole␈αsequence.
␈β∧d␈↓ ↓`␈ε"b)␈↓ α␈ε"T␈α⎇ak␈α␈e␈αalternate␈αterms␈αof␈αthe␈αsequence,␈αstarting␈αwith␈αthe␈α|rst.
␈β¬⊂␈↓ ↓d␈ε"c)␈↓ α␈ε"T␈α⎇ak␈α␈e␈αthe␈αterms␈αof␈αthe␈αsequence␈αfollo␈α␈wing␈αa␈αzero.
␈β¬;␈↓ ↓`␈ε"d)␈↓ α␈ε"T␈α⎇ak␈α␈e␈αthe␈αterms␈αof␈αthe␈αsequence␈αfollo␈α␈wing␈αa␈αone.
␈β¬u␈↓ ε)␈ε%1
␈β¬z␈↓ α␈ε"No␈α␈w␈αa␈αsequence␈↓ ∧⊃␈ε(X␈↓ ∧>␈ε",␈↓ ∧T␈ε".␈αε.␈αε.␈↓ ¬∧␈ε",␈↓ ¬~␈ε(X␈↓ ¬S␈ε"is␈α(4,␈↓ ε=␈ε")-random␈αif:
␈βεε␈↓ ∧/␈ε%1␈↓ ¬8␈ε%8
␈βε␈↓ ε)␈ε%8
␈βε∞␈↓ ε)␈∧ε∞ε)α⊂
␈βε4␈↓ α↑␈ε%1␈↓ ¬.␈ε%1␈↓ επ␈ε%1
␈βε9␈↓ ↓H␈ε"by␈α∃(a),␈↓ αP␈ε6j␈↓ αr␈ε"(␈↓ α}␈ε(X␈↓ β3␈ε"+␈↓ β←␈ε6↓␈αε↓␈αε↓␈↓ ∧⊃␈ε"+␈↓ ∧=␈ε(X␈↓ ∧j␈ε")␈ε6␈αλ␈␈↓ ¬A␈ε6j␈α
∀␈↓ ε~␈ε",␈αi.e.,␈αif␈αthere␈αare␈α3,␈α4,␈αor␈α5␈αones;
␈βεE␈↓ β≤␈ε%1␈↓ ∧[␈ε%8
␈βεJ␈↓ α↑␈ε%8␈↓ ¬.␈ε%2␈↓ επ␈ε%8
␈βεM␈↓ α↑␈∧εMα↑α⊂␈↓ ¬.␈∧εM¬.α⊂␈↓ επ␈∧εMεπα⊂
␈βεi␈↓ α↑␈ε%1␈↓ ε→␈ε%1␈↓ εw␈ε%1
␈βεn␈↓ ↓H␈ε"by␈α∀(b),␈↓ αP␈ε6j␈↓ αr␈ε"(␈↓ α}␈ε(X␈↓ β4␈ε"+␈↓ βa␈ε(X␈↓ ∧↔␈ε"+␈↓ ∧C␈ε(X␈↓ ∧y␈ε"+␈↓ ¬&␈ε(X␈↓ ¬S␈ε")␈ε6␈α	␈␈↓ ε,␈ε6j␈α∀␈↓ π
␈ε",␈α
i.e.,␈α∞if␈α
there␈α
are␈α∞t␈α␈w␈α␈o␈α
ones␈α
in␈α
odd-
␈βεz␈↓ β≤␈ε%1␈↓ β␈␈ε%3␈↓ ∧a␈ε%5␈↓ ¬D␈ε%7
␈βε␈␈↓ α↑␈ε%4␈↓ ε→␈ε%2␈↓ εw␈ε%8
␈βπα␈↓ α↑␈∧παα↑α⊂␈↓ ε→␈∧παε→α⊂␈↓ εw␈∧παεwα⊂
␈βπ→␈↓ αP␈ε"n␈α␈um␈α␈bered␈αpositions;
␈βπN␈↓ ↓H␈ε"by␈α⊗(c),␈↓ αP␈ε"there␈αare␈αthree␈αpossibilities␈αdepending␈αon␈αh␈α↓o␈α␈w␈αman␈α␈y␈αzeros␈αoccupy␈αposi-
␈βπy␈↓ αP␈ε"tions␈↓ β*␈ε(X␈↓ βW␈ε",␈↓ βp␈ε".␈αε.␈αε.␈↓ ∧ ␈ε",␈↓ ∧8␈ε(X␈↓ ∧e␈ε":␈α⊂if␈α∞there␈α∞are␈α
2␈α∞or␈α∞3␈α∞zeros␈α∞here,␈α∞there␈α∞is␈α∞n␈α↓o␈α
condition
␈βλε␈↓ βH␈ε%1␈↓ ∧V␈ε%7
␈βλ$␈↓ αP␈ε"to␈α⊂test␈α∂(since␈ε(␈α⊂n␈ε"␈α∂=␈α⊂4);␈α⊃if␈α⊂there␈α∂are␈α∂4␈α⊂zeros,␈α⊂they␈α⊂m␈α␈ust␈α∂respectiv␈α␈ely␈α∂be
␈βλP␈↓ αP␈ε"follo␈α␈w␈α␈ed␈αby␈αt␈α␈w␈α␈o␈αzeros␈αand␈αt␈α␈w␈α␈o␈αones;␈αand␈αif␈αthere␈αare␈α5␈αzeros,␈αthey␈αm␈α␈ust
␈βλ{␈↓ αP␈ε"respectiv␈α␈ely␈αbe␈αfollo␈α␈w␈α␈ed␈αby␈αt␈α␈w␈α␈o␈αor␈αthree␈αzeros;
␈β	0␈↓ ↓H␈ε"by␈α∀(d),␈↓ αP␈ε"w␈α␈e␈αget␈αconditions␈αsimilar␈αto␈αth␈α↓ose␈αimplied␈αby␈α(c).
␈β	j␈↓ ↓␈ε%1
␈β	o␈↓ α␈ε"It␈α
turns␈αout␈αthat␈α
only␈αthe␈α
follo␈α␈wing␈αbinary␈α
sequences␈αof␈α
length␈α8␈α
are␈α(4,␈↓ ∀␈ε")-
␈β
␈↓ ↓␈ε%8
␈β
β␈↓ ↓␈∧
β↓α⊂
␈β
~␈↓ ↓H␈ε"random␈αwith␈αrespect␈αto␈αthese␈αrules:
␈β
c␈↓ β.␈ε"00001011␈↓ ¬ε␈ε"00101001␈↓ ε↑␈ε"01001110␈↓ λ6␈ε"01101000
␈β∞␈↓ β.␈ε"00011010␈↓ ¬ε␈ε"00101100␈↓ ε↑␈ε"01011011␈↓ λ6␈ε"01101100
␈β9␈↓ β.␈ε"00011011␈↓ ¬ε␈ε"00110010␈↓ ε↑␈ε"01011110␈↓ λ6␈ε"01101101
␈βe␈↓ β.␈ε"00100011␈↓ ¬ε␈ε"00110011␈↓ ε↑␈ε"01100010␈↓ λ6␈ε"01110010
␈β⊂␈↓ β.␈ε"00100110␈↓ ¬ε␈ε"00110110␈↓ ε↑␈ε"01100011␈↓ λ6␈ε"01110110
␈β;␈↓ β.␈ε"00100111␈↓ ¬ε␈ε"00111001␈↓ ε↑␈ε"01100110
␈β␈␈↓ ↓H␈ε"plus␈αth␈α↓ose␈αobtained␈αby␈αin␈α␈terchanging␈α0␈αand␈α1␈αconsisten␈α␈tly.
␈β
*␈↓ α␈ε"It␈α	is␈α	clear␈α	that␈α	w␈α␈e␈α	could␈αλmak␈α␈e␈α	the␈α	set␈α	of␈α	algorithms␈α	so␈α	large␈α	that␈α	n␈α↓o␈αλsequences
␈β
U␈↓ ↓H␈ε"satisfy␈α	the␈α	de|nition,␈α
when␈ε(␈α	n␈ε"␈α	and␈ε(␈α	∂␈ε"␈α	are␈α	reasonably␈α	small.␈α∂A.␈α	N.␈↓ 	≡␈ε"Kolom␈α↓ogoro␈α␈v␈α	has
␈β∞␈↓ ↓H␈ε"pro␈α␈v␈α␈ed␈α
that␈αan␈α(␈ε(n␈ε"␈α␈,␈ε(␈αε∂␈ε")-random␈αbinary␈α
sequence␈ε/␈αwill␈ε"␈αalw␈α␈a␈α␈ys␈αexist,␈α
for␈αan␈α␈y␈αgiv␈α␈en␈↓ 
␈␈ε(N␈↓ "␈ε",
␈β∞+␈↓ ↓H␈ε"if␈αthe␈αn␈α␈um␈α␈ber␈αof␈αalgorithms␈αin␈ε2␈αA␈ε"␈αdoes␈αn␈α↓ot␈αexceed
␈β∞t␈↓ ¬←␈ε"1
␈β∞␈␈↓ ε4␈ε'2
␈β∂∧␈↓ ε∧␈ε%2␈ε+n␈↓ ε(␈ε+∂␈↓ εA␈ε%(␈α↓1␈ε9␈␈ε+␈α␈∂␈ε%␈α↓)
␈β∂␈↓ ¬u␈ε(e␈↓ π∂␈ε".␈↓ 
p␈ε"(35)
␈β∂ ␈↓ ¬←␈∧∂ ¬←α∩
␈β∂&␈↓ ¬←␈ε"2
␈β∂i␈↓ ↓H␈ε"This␈α
result␈αis␈αn␈α↓ot␈αnearly␈α
strong␈αen␈α↓ough␈αto␈αsh␈α↓o␈α␈w␈α
that␈αsequences␈αsatisfying␈α
De|ni-
␈β⊂∃␈↓ ↓H␈ε"tion␈αλQ1␈αλwill␈αλexist,␈α	but␈αλthe␈αλlatter␈α	can␈αλbe␈αλconstructed␈αλe}cien␈α␈tly␈αλusing␈αλthe␈αλprocedure
␈β⊂@␈↓ ↓H␈ε"of␈↓ ↓r␈ε"Rees␈αin␈αexercise␈α3.2.2↑21.
␈β⊂k␈↓ α␈ε"Still␈α∂an␈α↓other␈α⊂in␈α␈teresting␈α∂approach␈α⊂to␈α∂a␈α⊂de|nition␈α∂of␈α⊂randomness␈α∂has␈α∂been
␈β⊃⊗␈↓ ↓H␈ε"tak␈α␈en␈αby␈α
Per␈↓ β"␈ε"Martin-L␈↓ ∧2␈ε"∪␈↓ ∧2␈ε"o␈↓ ∧D␈ε"f␈α
[␈ε/Information␈αand␈α
Con␈α␈trol␈ε2␈α
9␈ε"␈α
(1966),␈α602↑619].␈α∪Giv␈α␈en␈αa
␈β∪(

␈β↓U␈↓ ↓H␈ε"164␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα"␈↓ ↓H␈ε"|nite␈ε(␈α	b␈ε"␈α↓-ary␈α
sequence␈↓ ∧␈ε(X␈↓ ∧:␈ε",␈↓ ∧N␈ε".␈αε.␈αε.␈↓ ∧}␈ε",␈↓ ¬∩␈ε(X␈↓ ¬L␈ε",␈α
let␈↓ ε∩␈ε(l␈↓ ε≥␈ε"(␈↓ ε)␈ε(X␈↓ εW␈ε",␈↓ εg␈ε".␈αε.␈αε.␈↓ π↔␈ε",␈↓ π'␈ε(X␈↓ π`␈ε")␈α
be␈α
the␈α
length␈α
of␈α
the␈α	sh␈α↓ortest
␈βα/␈↓ ∧*␈ε%1␈↓ ¬0␈ε+N␈↓ εG␈ε%1␈↓ πE␈ε+N
␈βαM␈↓ ↓H␈ε"T␈α⎇uring␈α	machine␈α	program␈α	that␈αλgenerates␈α	this␈α	sequence.␈α∂(F␈α⎇or␈α	a␈α	de|nition␈αλof␈α	T␈α⎇uring
␈βαy␈↓ ↓H␈ε"machines,␈αsee␈αChapter␈α11;␈αalternativ␈α␈ely,␈αw␈α␈e␈αcould␈αuse␈αcertain␈αclasses␈αof␈αe{ectiv␈α␈e
␈ββ$␈↓ ↓H␈ε"algorithms,␈α∂such␈α∂as␈α∂th␈α↓ose␈α∂de|ned␈α∂in␈α∂exercise␈α⊂1.1↑8.)␈α≡Then␈↓ 	
␈ε(l␈↓ 	⊗␈ε"(␈↓ 	"␈ε(X␈↓ 	O␈ε",␈↓ 	←␈ε".␈αε.␈αε.␈↓ 
∂␈ε",␈↓ 
∨␈ε(X␈↓ 
X␈ε")␈α∂is␈α∂a
␈ββ0␈↓ 	@␈ε%1␈↓ 
=␈ε+N
␈ββO␈↓ ↓H␈ε"measure␈α
of␈α
the␈α∞\patternlessness"␈α
of␈α
the␈α
sequence,␈α∞and␈α
w␈α␈e␈α∞ma␈α␈y␈α
equate␈α
this␈α
idea
␈ββz␈↓ ↓H␈ε"with␈αrandomness.␈α∂The␈αsequences␈αof␈αlength␈↓ εj␈ε(N␈↓ π_␈ε"that␈α
maximize␈↓ 	¬␈ε(l␈↓ 	⊃␈ε"(␈↓ 	≥␈ε(X␈↓ 	J␈ε",␈↓ 	Z␈ε".␈αε.␈αε.␈↓ 

␈ε",␈↓ 
~␈ε(X␈↓ 
T␈ε")␈α
ma␈α␈y
␈β∧π␈↓ 	;␈ε%1␈↓ 
8␈ε+N
␈β∧%␈↓ ↓H␈ε"be␈α
called␈αrandom.␈α∃(F␈α⎇rom␈α
the␈αstandpoin␈α␈t␈αof␈α
practical␈αrandom␈αn␈α␈um␈α␈ber␈α
generation
␈β∧Q␈↓ ↓H␈ε"by␈α∞computer,␈α∂this␈α∞is,␈α∂of␈α∞course,␈α∂the␈α∞w␈α␈orst␈α∞de|nition␈α∂of␈α∞\randomness"␈α∞that␈α∞can
␈β∧|␈↓ ↓H␈ε"be␈αimagined!)
␈β¬'␈↓ α␈ε"Essen␈α␈tially␈α	the␈α	same␈α	de|nition␈α	of␈α	randomness␈α	w␈α␈as␈α	independen␈α␈tly␈α	giv␈α␈en␈α	by␈α	G.
␈β¬R␈↓ ↓H␈ε"Chaitin␈αat␈αabout␈α
the␈αsame␈αtime;␈α
see␈ε/␈αJA␈α␈CM␈ε2␈α16␈ε"␈α
(1969),␈α145↑159.␈α⊃It␈αis␈αin␈α␈teresting
␈β¬⎇␈↓ ↓H␈ε"to␈α
n␈α↓ote␈α
that␈α
ev␈α␈en␈α
th␈α↓ough␈α
this␈α
de|nition␈α
mak␈α␈es␈αn␈α↓o␈α
reference␈α
to␈α
equidistribution
␈βε)␈↓ ↓H␈ε"properties␈αλas␈α	our␈α	other␈αλde|nitions␈α	ha␈α␈v␈α␈e,␈α	Martin-L␈↓ πA␈ε"∪␈↓ πA␈ε"o␈↓ πS␈ε"f␈α	and␈αλChaitin␈α	ha␈α␈v␈α␈e␈α	pro␈α␈v␈α␈ed␈αλthat
␈βεT␈↓ ↓H␈ε"random␈α	sequences␈α	of␈α
this␈α	t␈α␈ype␈α
also␈α	ha␈α␈v␈α␈e␈α	the␈α
expected␈α	equidistribution␈α	properties.
␈βε␈␈↓ ↓H␈ε"In␈α
fact,␈α
Martin-L␈↓ βR␈ε"∪␈↓ βR␈ε"o␈↓ βd␈ε"f␈αhas␈α
dem␈α↓onstrated␈α
that␈α
such␈αsequences␈α
satisfy␈ε/␈α
all␈ε"␈α
computable
␈βπ*␈↓ ↓H␈ε"statistical␈αtests␈αfor␈αrandomness,␈αin␈αan␈αappropriate␈αsense.
␈βπU␈↓ α␈ε"F␈α⎇or␈α∂further␈α∞dev␈α␈elopmen␈α␈ts␈α∂in␈α∞the␈α∂de|nition␈α∞of␈α∂random␈α∂|nite␈α∞sequences,␈α∂see
␈βλ↓␈↓ ↓H␈ε"A.␈α
K.␈↓ α-␈ε"Zv␈α␈onkin␈α∞and␈α
L.␈α∞A.␈↓ ∧d␈ε"Levin,␈ε/␈α∞Uspekhi␈α
Mat.␈α∞Nauk␈ε2␈α∞25␈ε",6␈α
(No␈α␈v.␈α∃1970),␈α∞85↑127
␈βλ,␈↓ ↓H␈ε"[English␈αtranslation␈α
in␈ε/␈α
Russian␈αMath.␈α
Surv␈α␈eys␈ε2␈α
25␈ε",6␈α
(No␈α␈v.␈α∩1970),␈α
83↑124];␈α
L.␈αA.
␈βλW␈↓ ↓H␈ε"Levin,␈ε/␈αDoklady␈αAk␈α⎇ad.␈αNauk␈αSSSR␈ε2␈α212␈ε"␈α(1973),␈α548↑550.
␈β	␈↓ ↓H␈ε2F.␈α⊃Summary,␈α∩history,␈α∩and␈α⊃bibliograph␈α␈y.␈ε"␈α"We␈α⊃ha␈α␈v␈α␈e␈α⊃de|ned␈α⊃sev␈α␈eral␈α⊃degrees␈α⊃of
␈β	6␈↓ ↓H␈ε"randomness␈αthat␈αa␈αsequence␈αmigh␈α␈t␈αpossess.
␈β	b␈↓ α␈ε"An␈α∪in|nite␈α∩sequence␈α∪that␈α∪is␈↓ ¬g␈ε61␈ε"-distributed␈α∪satis|es␈α∪a␈α∩great␈α∪man␈α␈y␈α∩useful
␈β

␈↓ ↓H␈ε"properties␈α⊂that␈α⊂are␈α⊂expected␈α⊂of␈α⊂random␈α⊂sequences,␈α⊃and␈α⊂there␈α⊂is␈α⊂a␈α⊂rich␈α⊂theory
␈β
8␈↓ ↓H␈ε"concerning␈ε6␈α1␈ε"-distributed␈αsequences.␈α~(The␈αexercises␈α
that␈αfollo␈α␈w␈αdev␈α␈elop␈αsev␈α␈eral
␈β
c␈↓ ↓H␈ε"importan␈α␈t␈αproperties␈α
of␈ε6␈α
1␈ε"-distributed␈α
sequences␈αthat␈α
ha␈α␈v␈α␈e␈α
n␈α↓ot␈α
been␈αmen␈α␈tioned
␈β∞␈↓ ↓H␈ε"in␈αλthe␈α	text.)␈α
De|nition␈αλR1␈α	is␈αλtherefore␈α	an␈αλappropriate␈α	basis␈αλfor␈α	theoretical␈αλstudies
␈β:␈↓ ↓H␈ε"of␈αrandomness.
␈βe␈↓ α␈ε"The␈αconcept␈αof␈α
an␈ε6␈α1␈ε"-distributed␈ε(␈αb␈ε"-ary␈αsequence␈α
w␈α␈as␈αin␈α␈troduced␈αin␈α1909␈αby
␈β⊂␈↓ ↓H␈ε"Emile␈↓ α+␈ε"Borel.␈α∂He␈α	essen␈α␈tially␈α	de|ned␈α	the␈α	concept␈α	of␈α	an␈α	(␈ε(m␈ε",␈↓ λ9␈ε(k␈↓ λM␈ε")-distributed␈α	sequence,
␈β;␈↓ ↓H␈ε"and␈αsh␈α↓o␈α␈w␈α␈ed␈αthat␈αthe␈ε(␈αb␈ε"-ary␈αrepresen␈α␈tations␈αof␈αalm␈α↓ost␈αall␈αreal␈αn␈α␈um␈α␈bers␈αare␈α(␈ε(m␈ε"␈α␈,␈↓ ␈ε(k␈↓ ∀␈ε")-
␈βf␈↓ ↓H␈ε"distributed␈α⊂for␈α⊃all␈ε(␈α⊂m␈ε"␈α⊂and␈↓ ∧r␈ε(k␈↓ ¬ε␈ε".␈α≥He␈α⊃called␈α⊂such␈α⊃n␈α␈um␈α␈bers␈ε/␈α⊂n␈α↓ormal␈↓ 	B␈ε"to␈α⊂base␈ε(␈α⊃b␈ε".␈α≥An
␈β
∩␈↓ ↓H␈ε"excellen␈α␈t␈αdiscussion␈α
of␈α
this␈αtopic␈α
appears␈αin␈α
his␈α
w␈α␈ell-kn␈α↓o␈α␈wn␈αbo␈α↓ok,␈ε/␈α
Le␈↓ 	⎇␈ε/_␈↓ 
β␈ε/c␈↓ 
∪␈ε/ons␈α
sur␈αla
␈β
=␈↓ ↓H␈ε/th␈↓ ↓i␈ε/∞␈↓ ↓j␈ε/e␈↓ ↓z␈ε/orie␈αdes␈αfonctions␈ε"␈α(2nd␈αed.,␈α1914),␈α182↑216.
␈β
h␈↓ α␈ε"The␈α∩n␈α↓otion␈α∩of␈α∩an␈ε6␈α∩1␈ε"-distributed␈α∩sequence␈α∩of␈ε/␈α∩real␈ε"␈α∩n␈α␈um␈α␈bers,␈α∪also␈α∩called␈α∩a
␈β∞∪␈↓ ↓H␈ε/completely␈αequidistributed␈αsequence␈ε",␈α|rst␈αappeared␈αin␈αa␈αn␈α↓ote␈αby␈αN.␈αM.␈αKorobo␈α␈v
␈β∞>␈↓ ↓H␈ε"in␈ε/␈α⊂Doklady␈α⊂Ak␈α⎇ad.␈α⊃Nauk␈α⊂SSSR␈ε2␈α⊃62␈ε"␈α⊂(1948),␈α∩21↑22.␈↓ πw␈ε"Korobo␈α␈v␈α⊂and␈α⊃sev␈α␈eral␈α⊂of␈α⊂his
␈β∞j␈↓ ↓H␈ε"colleagues␈α
dev␈α␈eloped␈αthe␈αtheory␈αof␈α
such␈αsequences␈αquite␈αextensiv␈α␈ely␈α
in␈αa␈αseries␈α
of
␈β∂∃␈↓ ↓H␈ε"papers␈α
during␈α
the␈α
1950s.␈α∀Completely␈α∞equidistributed␈α
sequences␈α
w␈α␈ere␈α
independ-
␈β∂@␈↓ ↓H␈ε"en␈α␈tly␈α∂studied␈α∂by␈α⊂Joel␈α∂N.␈↓ ∧`␈ε"F␈α⎇ranklin,␈ε/␈α⊂Math.␈α∂Comp.␈ε2␈α⊂17␈ε"␈α∂(1963),␈α⊂28↑59,␈α⊃in␈α∂a␈α∂paper
␈β∂k␈↓ ↓H␈ε"that␈α
is␈α∞particularly␈α∞n␈α↓otew␈α␈orth␈α␈y␈α∞because␈α∞it␈α∞w␈α␈as␈α
inspired␈α∞by␈α∞the␈α∞problem␈α∞of␈α
ran-
␈β⊂⊗␈↓ ↓H␈ε"dom␈α⊂n␈α␈um␈α␈ber␈α⊂generation.␈α≤The␈α⊂bo␈α↓ok␈ε/␈α⊂Uniform␈α⊂Distribution␈α⊂of␈α⊂Sequences␈ε"␈α⊂by␈α⊂L.
␈β⊂B␈↓ ↓H␈ε"Kuipers␈αand␈αH.␈↓ βA␈ε"Niederreiter␈α(New␈αY␈α⎇ork:␈αWiley,␈α1974)␈αis␈αan␈αextraordinarily␈αcom-
␈β⊂m␈↓ ↓H␈ε"plete␈α∂source␈α∂of␈α∂information␈α∂about␈α⊂the␈α∂rich␈α∂mathematical␈α∂literature␈α∂concerning
␈β⊃_␈↓ ↓H␈ε(k␈↓ ↓\␈ε"-distributed␈↓ β ␈ε"sequences␈αof␈αall␈αkinds.
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"165
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα$␈↓ α␈ε"We␈αλha␈α␈v␈α␈e␈αλseen,␈α	h␈α↓o␈α␈w␈α␈ev␈α␈er,␈α	that␈ε6␈α	1␈ε"-distributed␈αλsequences␈αλneed␈αλn␈α↓ot␈αλbe␈αλsu}cien␈α␈tly
␈βαO␈↓ ↓H␈ε"haphazard␈α∞to␈α∞qualify␈α
completely␈α∞as␈α∞\random."␈α⊗Three␈α∞de|nitions,␈α∂R4,␈α∞R5,␈α∞and
␈βαz␈↓ ↓H␈ε"R6,␈αw␈α␈ere␈α
form␈α␈ulated␈αabo␈α␈v␈α␈e␈α
to␈αpro␈α␈vide␈α
the␈α
additional␈αconditions;␈α
and␈αDe|nition
␈ββ&␈↓ ↓H␈ε"R6,␈α∂in␈α∞particular,␈α∂seems␈α∞to␈α∞be␈α∂an␈α∞appropriate␈α∞w␈α␈a␈α␈y␈α∂to␈α∞de|ne␈α∞the␈α∂concept␈α∞of␈α∞an
␈ββQ␈↓ ↓H␈ε"in|nite␈α
random␈α∞sequence.␈α∀It␈α∞is␈α
a␈α∞precise,␈α
quan␈α␈titativ␈α␈e␈α∞statemen␈α␈t␈α
that␈α∞ma␈α␈y␈α
w␈α␈ell
␈ββ|␈↓ ↓H␈ε"coincide␈αwith␈αthe␈αin␈α␈tuitiv␈α␈e␈αidea␈αof␈αtrue␈αrandomness.
␈β∧)␈↓ α␈ε"Historically,␈α∂the␈α∂dev␈α␈elopmen␈α␈t␈α∞of␈α∂these␈α∞de|nitions␈α∞w␈α␈as␈α∂primarily␈α∞in⎇uenced
␈β∧T␈↓ ↓H␈ε"by␈αλR.␈↓ α&␈ε"v␈α␈on␈α	Mises'␈α	quest␈α	for␈α	a␈αλgo␈α↓od␈α	de|nition␈α	of␈α	\probabilit␈α␈y."␈α∂In␈ε/␈α	Math.␈αλZeitschri$
␈β∧␈␈↓ ↓H␈ε25␈ε"␈α	(1919),␈α	52↑99,␈α
v␈α␈on␈α	Mises␈α	proposed␈α	a␈α	de|nition␈α	similar␈α	in␈α	spirit␈α	to␈α	De|nition␈α	R5,
␈β¬*␈↓ ↓H␈ε"alth␈α↓ough␈αstated␈αto␈α↓o␈αstrongly␈α(lik␈α␈e␈αour␈αDe|nition␈αR3)␈αso␈αthat␈αn␈α↓o␈αsequences␈αsatis-
␈β¬U␈↓ ↓H␈ε"fying␈αthe␈αconditions␈αcould␈αpossibly␈αexist.␈α⊂Man␈α␈y␈αpeople␈αn␈α↓oticed␈αthis␈αdiscrepancy,
␈βε↓␈↓ ↓H␈ε"and␈α∞A.␈α∞H.␈↓ αt␈ε"Copeland␈α∞[␈ε/Amer.␈α∂J.␈α∞Math.␈ε2␈α∞50␈ε"␈α∞(1928),␈α∂535↑552]␈α∞suggested␈α∞w␈α␈eak␈α␈ening
␈βε,␈↓ ↓H␈ε"v␈α␈on␈α∞Mises'␈α∂de|nition␈α∂by␈α∂substituting␈α∂what␈α∞he␈α∂called␈α∂\␈↓ λ+␈ε"admissible␈α∂n␈α␈um␈α␈bers"␈α∞(or
␈βεW␈↓ ↓H␈ε"Bern␈α↓oulli␈α
sequences).␈α∃These␈α
are␈α∞equiv␈α⎇alen␈α␈t␈α
to␈ε6␈α∞1␈ε"-distributed␈↓ 	
␈ε"[␈↓ 	≥␈ε"0,␈αε1)␈α∞sequences␈α
in
␈βπα␈↓ ↓H␈ε"which␈α⊂all␈α⊂en␈α␈tries␈↓ βa␈ε(U␈↓ ∧≤␈ε"ha␈α␈v␈α␈e␈α⊂been␈α⊂replaced␈α⊂by␈α⊂1␈α⊂if␈↓ π\␈ε(U␈↓ λ_␈ε"<␈ε(␈α⊃p␈ε"␈α⊂or␈α⊂by␈α⊂0␈α⊂if␈↓ 
∨␈ε(U␈↓ 
[␈ε6∃␈ε(␈α⊂p␈ε",
␈βπ∂␈↓ βy␈ε+n␈↓ πu␈ε+n␈↓ 
7␈ε+n
␈βπ-␈↓ ↓H␈ε"for␈αa␈αgiv␈α␈en␈α
probabilit␈α␈y␈ε(␈αp␈ε".␈α∩Th␈α␈us␈αCopeland␈αw␈α␈as␈α
essen␈α␈tially␈αsuggesting␈α
a␈αreturn␈αto
␈βπY␈↓ ↓H␈ε"De|nition␈αR1.␈α∩Then␈αAbraham␈↓ ¬7␈ε"Wald␈α
sh␈α↓o␈α␈w␈α␈ed␈αthat␈α
it␈αis␈α
n␈α↓ot␈αnecessary␈α
to␈αw␈α␈eak␈α␈en
␈βλ∧␈↓ ↓H␈ε"v␈α␈on␈α
Mises'␈α
de|nition␈α
so␈αdrastically,␈α
and␈α
he␈αproposed␈α
substituting␈α
a␈α
coun␈α␈table␈α
set
␈βλ/␈↓ ↓H␈ε"of␈αλsubsequence␈α	rules.␈α∂In␈α	an␈α	importan␈α␈t␈αλpaper␈α	[␈ε/Ergebnisse␈α	eines␈α	math.␈αλKolloquiums
␈βλZ␈↓ ↓H␈ε28␈ε"␈α∩(Vienna,␈α∀1937),␈α∀38↑72],␈α∪Wald␈α∪essen␈α␈tially␈α∩pro␈α␈v␈α␈ed␈α∩Theorem␈α∩W␈↓ 	M␈ε",␈α∀alth␈α↓ough␈α∩he
␈β	¬␈↓ ↓H␈ε"made␈α⊂the␈α⊂erroneous␈α⊂assertion␈α⊂that␈α⊃the␈α⊂sequence␈α⊂constructed␈α⊂by␈α⊂Algorithm␈α⊂W
␈β	1␈↓ ↓H␈ε"also␈α∂satis|es␈α⊂the␈α⊂stronger␈α∂condition␈α⊂that␈↓ ε`␈ε"P␈↓ εx␈ε"r␈↓ πε␈ε"(␈↓ π∩␈ε(U␈↓ πN␈ε62␈ε(␈α⊂A␈ε")␈α⊂=␈↓ λ`␈ε"measure␈αof␈↓ 
⊗␈ε(A␈ε",␈α⊃for␈α∂all
␈β	=␈↓ π+␈ε+n
␈β	\␈↓ ↓H␈ε"Lebesgue␈α
measurable␈ε(␈α
A␈ε6␈α
∩␈↓ ∧n␈ε"[␈↓ ∧}␈ε"0,␈αε1).␈α∂We␈α
ha␈α␈v␈α␈e␈α
observ␈α␈ed␈α
that␈α
n␈α↓o␈α
sequence␈α
can␈α
satisfy
␈β
π␈↓ ↓H␈ε"this␈αpropert␈α␈y.
␈β
4␈↓ α␈ε"The␈α⊃concept␈α⊃of␈α⊃\␈↓ ∧%␈ε"computabilit␈α␈y"␈α⊃w␈α␈as␈α⊃still␈α⊃v␈α␈ery␈α⊃m␈α␈uch␈α⊃in␈α⊃its␈α⊂infancy␈α⊃when
␈β
←␈↓ ↓H␈ε"Wald␈α
wrote␈α∞his␈α
paper,␈α∂and␈α
A.␈↓ ¬2␈ε"Ch␈α␈urch␈α
[␈ε/Bull.␈α∞Amer.␈α∞Math.␈α
Soc.␈ε2␈α∞47␈ε"␈α
(1940),␈α∞130↑
␈β
␈↓ ↓H␈ε"135]␈α∩sh␈α↓o␈α␈w␈α␈ed␈α⊃h␈α↓o␈α␈w␈α∩the␈α∩precise␈α∩n␈α↓otion␈α∩of␈α∩\␈↓ εg␈ε"e{ectiv␈α␈e␈α∩algorithm"␈α∩could␈α∩be␈α⊃added
␈β5␈↓ ↓H␈ε"to␈α⊂Wald's␈α⊂theory␈α⊃to␈α⊂mak␈α␈e␈α⊂his␈α⊃de|nitions␈α⊂completely␈α⊂rigorous.␈α≥The␈α⊂extension
␈β`␈↓ ↓H␈ε"to␈α∂De|nition␈α∂R6␈α⊂w␈α␈as␈α∂due␈α⊂essen␈α␈tially␈α∂to␈α∂A.␈α⊂N.␈↓ π7␈ε"Kolm␈α↓ogoro␈α␈v␈α⊂[␈ε/Sankh␈α␈y␈↓ 
∧␈ε/∩␈↓ 
∧␈ε/a␈↓ 
%␈ε"series␈α∂A,
␈β␈↓ ↓H␈ε225␈ε"␈α⊂(1963),␈α∪369↑376],␈α∩wh␈α↓o␈α⊂proposed␈α⊃De|nition␈α⊃Q2␈α⊃for␈α⊃|nite␈α⊃sequences␈α⊃in␈α⊂that
␈β7␈↓ ↓H␈ε"same␈α
paper.␈α⊗An␈α↓other␈α
de|nition␈α∞of␈α∞randomness␈α∞for␈α
|nite␈α∞sequences,␈α∞somewhere
␈βb␈↓ ↓H␈ε"\bet␈α␈w␈α␈een"␈α∞De|nitions␈α∞Q1␈α∞and␈α∞Q2,␈α∂had␈α∞been␈α∂form␈α␈ulated␈α∞man␈α␈y␈α∞y␈α␈ears␈α∞earlier␈α∞by
␈β

␈↓ ↓H␈ε"A.␈αS.␈↓ α"␈ε"Besico␈α␈vitch␈α[␈ε/Math.␈αZeitschri$␈ε2␈α39␈ε"␈α(1934),␈α146↑156].
␈β
:␈↓ α␈ε"The␈α∩publications␈α∪of␈α∩Ch␈α␈urch␈α∩and␈α∩Kolm␈α↓ogoro␈α␈v␈α∪considered␈α∩only␈α∩binary␈α∩se-
␈β
e␈↓ ↓H␈ε"quences␈α∞for␈α∂which␈↓ βt␈ε"P␈↓ ∧␈ε"r␈↓ ∧~␈ε"(␈↓ ∧&␈ε(X␈↓ ∧e␈ε"=␈α∞1)␈α∂=␈ε(␈α∞p␈ε"␈α∂for␈α∂a␈α∞giv␈α␈en␈α∂probabilit␈α␈y␈ε(␈α∂p␈ε".␈α_Our␈α∞discussion
␈β
q␈↓ ∧D␈ε+n
␈β∞⊂␈↓ ↓H␈ε"in␈αthis␈αsection␈αhas␈α
been␈αsligh␈α␈tly␈αm␈α↓ore␈α
general,␈αsince␈αa␈↓ λ≤␈ε"[␈↓ λ,␈ε"0,␈αε1)␈αsequence␈αessen␈α␈tially
␈β∞;␈↓ ↓H␈ε"represen␈α␈ts␈α	all␈ε(␈α
p␈ε"␈α	at␈α	once.␈α⊂The␈α	v␈α␈on␈α
Mises↑Wald↑Ch␈α␈urch␈α	de|nition␈α
has␈α	been␈α	re|ned
␈β∞f␈↓ ↓H␈ε"in␈α∂y␈α␈et␈α∞an␈α↓other␈α∂in␈α␈teresting␈α∂w␈α␈a␈α␈y␈α∂by␈α∂J.␈α∂V.␈↓ εO␈ε"Ho␈α␈w␈α␈ard,␈ε/␈α⊂Zeitschr.␈α∂f.␈α∂math.␈α∂Logik␈α∞und
␈β∂∩␈↓ ↓H␈ε/Grundlagen␈αd.␈αMath.␈ε2␈α21␈ε"␈α(1975),␈α215↑224.
␈β∂>␈↓ α␈ε"An␈α↓other␈α	importan␈α␈t␈αλcon␈α␈tribution␈α	w␈α␈as␈αλmade␈αλby␈α	Donald␈αλW.␈↓ 	β␈ε"Lo␈α␈v␈α␈eland␈αλ[␈ε/Zeitschr.
␈β∂i␈↓ ↓H␈ε/f.␈α⊂math.␈α⊃Logik␈α⊂und␈α⊃Grundlagen␈α⊂d.␈α⊃Math.␈ε2␈α⊂12␈ε"␈α⊃(1966),␈α⊃279↑294],␈α∩wh␈α↓o␈α⊂discussed
␈β⊂∃␈↓ ↓H␈ε"De|nitions␈α⊂R4,␈α⊃R5,␈α⊃R6,␈α∩and␈α⊂sev␈α␈eral␈α⊂in␈α␈termediate␈α⊂concepts.␈α≥Lo␈α␈v␈α␈eland␈α⊂pro␈α␈v␈α␈ed
␈β⊂@␈↓ ↓H␈ε"that␈α
there␈α
are␈αR5-random␈α
sequences␈α
that␈αdo␈α
n␈α↓ot␈α
satisfy␈α
R4,␈αthereby␈α
establishing
␈β⊂k␈↓ ↓H␈ε"the␈αneed␈αfor␈αa␈αstronger␈αde|nition␈αsuch␈αas␈αR6.␈α⊂In␈αfact,␈αhe␈αde|ned␈αa␈αrather␈αsimple
␈β⊃⊂␈↓ 	␈␈ε90
␈β⊃⊗␈↓ ↓H␈ε"perm␈α␈utation␈ε6␈αh␈↓ β ␈ε(f␈↓ β5␈ε"(␈ε(n␈ε")␈ε6i␈ε"␈αof␈αthe␈αn␈α↓onnegativ␈α␈e␈αin␈α␈tegers,␈αand␈αan␈αAlgorithm␈αW␈↓ 
∩␈ε"analogous
␈β∪(

␈β↓U␈↓ ↓H␈ε"166␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα≥␈↓ εB␈ε%1␈↓ λZ␈ε%1␈↓ 	9␈ε%1
␈βα!␈↓ ↓H␈ε"to␈α
Algorithm␈α
W␈↓ βF␈ε",␈α
such␈α
that␈↓ ∧␈␈∧α!∧␈α'␈↓ ∧␈␈ε"P␈↓ ¬↔␈ε"r␈↓ ¬&␈ε"(␈↓ ¬2␈ε(U␈↓ ε∂␈ε6∃␈↓ εV␈ε")␈ε6␈αλ␈␈↓ π↔␈ε"P␈↓ π/␈ε"r␈↓ π=␈ε"(␈↓ πI␈ε(U␈↓ λ&␈ε6∃␈↓ λm␈ε")␈ε6␈α∃␈↓ 	Y␈ε"for␈α
ev␈α␈ery␈α
R5-
␈βα.␈↓ ¬J␈ε+f␈↓ ¬[␈ε%(␈ε+␈α↓n␈ε%␈α␈)␈↓ πb␈ε+f␈↓ πs␈ε%(␈ε+␈α↓n␈ε%␈α␈)
␈βα2␈↓ εB␈ε%2␈↓ λZ␈ε%2␈↓ 	9␈ε%2
␈βα5␈↓ εB␈∧α5εBα⊂␈↓ λZ␈∧α5λZα⊂␈↓ 	9␈∧α5	9α⊂
␈βαC␈↓ π↔␈∧αCπ↔α'
␈βαF␈↓ πF␈ε90
␈βαL␈↓ ↓H␈ε"random␈α
sequence␈ε6␈αh␈↓ βi␈ε(U␈↓ ∧∀␈ε6i␈ε"␈αproduced␈αby␈α
Algorithm␈αW␈↓ πY␈ε"when␈α
it␈αis␈α
giv␈α␈en␈αan␈αin|nite␈α
set
␈βαY␈↓ ∧α␈ε+n
␈βαx␈↓ ↓H␈ε"of␈αsubsequence␈αrules␈↓ ∧∩␈ε6R␈↓ ∧>␈ε".
␈ββ∧␈↓ ∧.␈ε+k
␈ββ#␈↓ α␈ε"Alth␈α↓ough␈α	De|nition␈αλR6␈α	is␈αλin␈α␈tuitiv␈α␈ely␈α	m␈α␈uch␈αλstronger␈α	than␈αλR4,␈α	it␈α	is␈αλapparen␈α␈tly
␈ββN␈↓ ↓H␈ε"n␈α↓ot␈α
a␈α
simple␈αmatter␈α
to␈α
pro␈α␈v␈α␈e␈α
this␈αrigorously,␈α
and␈αfor␈α
sev␈α␈eral␈α
y␈α␈ears␈αit␈α
w␈α␈as␈α
an␈α
open
␈ββy␈↓ ↓H␈ε"question␈α
whether␈α
or␈α
n␈α↓ot␈α
R4␈α
implies␈α
R6.␈α∀Finally␈α
Th␈α↓omas␈↓ λ[␈ε"Herzog␈α
and␈α
James␈α
C.
␈β∧$␈↓ ↓H␈ε"Owings,␈αJr.,␈αdisco␈α␈v␈α␈ered␈αh␈α↓o␈α␈w␈αto␈αconstruct␈αa␈αlarge␈αfamily␈αof␈αsequences␈α
that␈αsatisfy
␈β∧P␈↓ ↓H␈ε"R4␈α	but␈α	n␈α↓ot␈α	R6.␈α∞[See␈ε/␈α	Zeitschr.␈α	f.␈α	math.␈α	Logik␈α	und␈α	Grundlagen␈α	d.␈α	Math.␈ε2␈α	22␈ε"␈α	(1976),
␈β∧{␈↓ ↓H␈ε"385↑389.]
␈β¬&␈↓ α␈ε"Kolm␈α↓ogoro␈α␈v␈α
wrote␈α
an␈α↓other␈αsigni|can␈α␈t␈α
paper␈α
[␈ε/Problem␈α␈y␈α
Pereda␈↓ 	[␈ε/⊂␈↓ 	\␈ε/c␈↓ 	l␈ε/i␈α
Informatsii
␈β¬Q␈↓ ↓H␈ε21␈ε"␈α(1965),␈α3↑11]␈αin␈αwhich␈αhe␈αconsidered␈αthe␈αproblem␈αof␈αde|ning␈αthe␈α\information
␈β¬⎇␈↓ ↓H␈ε"con␈α␈ten␈α␈t"␈α	of␈α
a␈α	sequence,␈α
and␈α
this␈α	w␈α␈ork␈α
led␈α	to␈↓ εu␈ε"Chaitin␈α	and␈↓ λ8␈ε"Martin-L␈↓ 	H␈ε"∪␈↓ 	H␈ε"o␈↓ 	Z␈ε"f's␈α	in␈α␈teresting
␈βε(␈↓ ↓H␈ε"de|nition␈α∂of␈α∂|nite␈α∂random␈α⊂sequences␈α∂via␈α∂\patternlessness."␈α∨[See␈ε/␈α∂IEEE␈α∂T␈α⎇rans.
␈βεS␈↓ ↓H␈ε2IT-14␈ε"␈α(1968),␈α662↑664.]
␈βε}␈↓ α␈ε"F␈α⎇or␈α
a␈αphilosophical␈αdiscussion␈α
of␈αrandom␈α
sequences,␈αsee␈α
K.␈αR.␈↓ 	h␈ε"Popper,␈ε/␈αThe
␈βπ)␈↓ ↓H␈ε/Logic␈αof␈αScien␈α␈ti|c␈αDisco␈α␈v␈α␈ery␈ε"␈α(London,␈α1959),␈αespecially␈αthe␈αin␈α␈teresting␈αconstruc-
␈βπU␈↓ ↓H␈ε"tion␈αon␈αpp.␈α162↑163,␈αwhich␈αhe␈α|rst␈αpublished␈αin␈α1934.
␈βλ␈↓ α␈ε"F␈α⎇urther␈αconnections␈α
bet␈α␈w␈α␈een␈αrandom␈α
sequences␈αand␈α
recursiv␈α␈e␈αfunction␈αthe-
␈βλ+␈↓ ↓H␈ε"ory␈α
ha␈α␈v␈α␈e␈α
been␈αexplored␈α
by␈αD.␈α
W.␈αLo␈α␈v␈α␈eland,␈ε/␈α
T␈α⎇rans.␈αAmer.␈α
Math.␈αSoc.␈ε2␈α
125␈ε"␈α
(1966),
␈βλV␈↓ ↓H␈ε"497↑510.␈α∨See␈α⊃also␈α⊂C.-P.␈↓ ∧c␈ε"Schn␈α↓orr␈α⊃[␈ε/Zeitschr.␈α⊃Wahr.␈α⊃v␈α␈erw.␈α⊃Geb.␈ε2␈α⊃14␈ε"␈α⊃(1969),␈α∩27↑
␈β	α␈↓ ↓H␈ε"35],␈α∪wh␈α↓o␈α∩found␈α∩strong␈α∩relations␈α∩bet␈α␈w␈α␈een␈α∩random␈α∩sequences␈α∩and␈α∩the␈α∩\␈↓ 
B␈ε"species
␈β	-␈↓ ↓H␈ε"of␈α⊂measure␈α⊂zero"␈α⊂de|ned␈α⊂by␈α⊂L.␈α⊂E.␈α⊂J.␈↓ ε/␈ε"Brouw␈α␈er␈α⊂in␈α⊂1919.␈α≤Schn␈α↓orr's␈α⊂subsequen␈α␈t
␈β	X␈↓ ↓H␈ε"bo␈α↓ok␈ε/␈α
Zuf␈↓ αX␈ε/∪␈↓ αX␈ε/a␈↓ αj␈ε/lligk␈α␈eit␈α∞und␈α∞Wahrscheinlichk␈α␈eit␈ε"␈α∞[␈ε/Lecture␈α
Notes␈α∞in␈α∞Math.␈ε2␈α∞218␈ε"␈α
(Berlin:
␈β
β␈↓ ↓H␈ε"Springer,␈α∞1971)]␈α
giv␈α␈es␈α
a␈α∞detailed␈α
treatmen␈α␈t␈α∞of␈α
the␈α∞en␈α␈tire␈α
subject␈α∞of␈α
randomness
␈β
.␈↓ ↓H␈ε"and␈αmak␈α␈es␈αan␈αexcellen␈α␈t␈αin␈α␈troduction␈αto␈αthe␈αev␈α␈er-gro␈α␈wing␈αadv␈α⎇anced␈αliterature␈αon
␈β
Z␈↓ ↓H␈ε"the␈αtopic.
␈β\␈↓ ↓H␈ε=E␈α␈XERCISES
␈β.␈↓ ↓g␈ε31.␈↓ α␈ε#[␈ε)10␈↓ α;␈ε#]␈α⊗Ca␈α␈n␈αa␈αp␈α␈eriod␈α␈i␈α↓c␈αse␈α␈que␈α␈nce␈αb␈α␈e␈↓ ¬o␈ε#eq␈α␈uidistrib␈α␈uted␈α␈?
␈βa␈↓ ↓g␈ε32.␈↓ α␈ε#[␈ε)10␈↓ α;␈ε#]␈α⊗Co␈α␈nside␈α␈r␈α∩th␈α␈e␈α⊃pe␈α␈ri␈α↓o␈α␈dic␈α⊃b␈α␈i␈α↓n␈α␈ary␈α⊃se␈α␈qu␈α␈ence␈α⊃0␈α␈,␈α∪0,␈α∪1,␈α∩1,␈α∪0,␈α∪0␈α␈,␈α∪1,␈α∪1,␈↓ 
⊃␈ε#.␈α¬.␈αε.␈↓ 
C␈ε#.␈α!Is␈α⊃it
␈β
	␈↓ ↓H␈ε#1␈α␈-distribu␈α␈ted␈↓ βα␈ε#?␈α∂Is␈αi␈α↓t␈α2␈α␈-distribu␈α␈ted?␈↓ ¬9␈ε#Is␈αit␈α3␈α␈-␈α↓d␈α␈istribu␈α␈ted?
␈β
;␈↓ ↓g␈ε33.␈↓ α␈ε#[␈ε)M2␈α␈2␈↓ α\␈ε#]␈α⊗Con␈α␈struct␈αa␈α
period␈α␈ic␈αtern␈α␈ary␈αse␈α␈que␈α␈nce␈αth␈α␈at␈αis␈α3-d␈α␈i␈α↓st␈α␈ri␈α↓b␈α␈ute␈α␈d.
␈β
h␈↓ ∧G␈ε:b␈↓ ∧R␈ε&lg␈↓ ∧h␈ε&(␈ε,␈α␈n␈ε&+␈α↓1␈α␈)␈ε:c
␈β
l␈↓ ↓;␈ε↓x␈↓ λi␈ε&1
␈β
n␈↓ ↓c␈ε34.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈22␈↓ αt␈ε#]␈α⊗Let␈↓ βP␈ε)U␈↓ ∧␈ε#=␈α
(␈↓ ∧6␈ε#2␈↓ ¬?␈ε#/3␈α␈)␈↓ ¬q␈ε#mo␈α␈d␈↓ ε5␈ε#1.␈α∂What␈αis␈↓ π\␈ε#P␈↓ πr␈ε#r␈↓ π␈␈ε#(␈↓ λ
␈ε)U␈↓ λ;␈ε#<␈↓ λ{␈ε#)?
␈β
z␈↓ βf␈ε,n␈↓ λ!␈ε,n
␈β
␈␈↓ λi␈ε&2
␈β∞↓␈↓ λi␈∧∞↓λiα∞
␈β∞∨␈↓ ∧U␈ε"(␈↓ ε1␈ε")␈↓ πβ␈ε"(␈↓ λF␈ε")␈↓ 	)␈ε"(␈↓ 	w␈ε")␈↓ 
I␈ε"(␈↓ ↔␈ε")
␈β∞!␈↓ ↓g␈ε35.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈14␈↓ αt␈ε#]␈α⊗Pro␈α␈v␈α}e␈αλth␈α␈at␈↓ ∧2␈ε#P␈↓ ∧H␈ε#r␈↓ ∧a␈ε)S␈↓ ∧x␈ε#(␈ε)n␈ε#)␈↓ ¬.␈ε#a␈α␈nd␈↓ ¬n␈ε)T␈↓ εε␈ε#(␈ε)n␈ε#)␈↓ ε=␈ε#+␈↓ ε←␈ε#P␈↓ εv␈ε#r␈↓ π∂␈ε)S␈↓ π&␈ε#(␈ε)n␈ε#)␈↓ π[␈ε#or␈↓ λ∧␈ε)T␈↓ λ≤␈ε#(␈ε)n␈ε#)␈↓ λ[␈ε#=␈↓ 	ε␈ε#P␈↓ 	≤␈ε#r␈↓ 	5␈ε)S␈↓ 	L␈ε#(␈ε)␈α↓n␈ε#)␈↓ 
∧␈ε#+␈↓ 
&␈ε#P␈↓ 
<␈ε#r␈↓ 
U␈ε)T␈↓ 
m␈ε#(␈ε)n␈ε#)␈↓ #␈ε#,
␈β∞I␈↓ ↓H␈ε#fo␈α␈r␈αan␈α}y␈α
t␈α␈w␈α␈o␈αsta␈α␈teme␈α␈n␈α␈ts␈↓ ∧∨␈ε)S␈↓ ∧6␈ε#(␈ε)n␈ε#),␈↓ ∧t␈ε)T␈↓ ¬␈ε#(␈ε)n␈ε#),␈αpro␈α}vide␈α␈d␈↓ εS␈ε#t␈α␈hat␈α
at␈α
l␈α↓e␈α␈ast␈α
three␈α
of␈α
the␈α
li␈α↓m␈α␈i␈α↓ts␈α
ex␈α␈i␈α↓st.␈α∞F␈α}o␈α␈r
␈β∞p␈↓ ↓H␈ε#e␈α␈xam␈α␈ple,␈αif␈αa␈α
sequ␈α␈enc␈α␈e␈αi␈α↓s␈α2␈α␈-distribu␈α␈ted,␈αw␈α␈e␈αw␈α␈ou␈α␈ld␈α|n␈α␈d␈αth␈α␈at
␈β∂@␈↓ ↓]␈ε#P␈↓ ↓s␈ε#r␈↓ α␈ε#(␈↓ α␈ε)u␈↓ α6␈ε7∀␈↓ α`␈ε)U␈↓ β⊃␈ε#<␈↓ β<␈ε)v␈↓ βe␈ε#or␈↓ ∧∞␈ε)u␈↓ ∧8␈ε7∀␈↓ ∧c␈ε)U␈↓ ¬<␈ε#<␈↓ ¬f␈ε)v␈↓ ε∧␈ε#)␈α
=␈↓ εC␈ε)v␈↓ εi␈ε7␈␈↓ π∩␈ε)u␈↓ π:␈ε#+␈↓ πc␈ε)v␈↓ λλ␈ε7␈␈↓ λ1␈ε)u␈↓ λZ␈ε7␈␈ε#␈απ(␈↓ 	∞␈ε)v␈↓ 	3␈ε7␈␈↓ 	\␈ε)u␈↓ 	⎇␈ε#)(␈↓ 
∪␈ε)v␈↓ 
9␈ε7␈␈↓ 
b␈ε)u␈↓ β␈ε#).
␈β∂K␈↓ α∨␈ε&1␈↓ αw␈ε,n␈↓ βL␈ε&1␈↓ ∧!␈ε&2␈↓ ∧y␈ε,n␈ε&+1␈↓ ¬v␈ε&2␈↓ εT␈ε&1␈↓ π%␈ε&1␈↓ πs␈ε&2␈↓ λE␈ε&2␈↓ 	≡␈ε&1␈↓ 	o␈ε&1␈↓ 
$␈ε&2␈↓ 
u␈ε&2
␈β⊂≠␈↓ ↓g␈ε36.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈23␈↓ αt␈ε#]␈α⊗Let␈↓ βM␈ε)S␈↓ βo␈ε#(␈ε)n␈ε#␈α↓),␈↓ ∧,␈ε)S␈↓ ∧N␈ε#(␈ε)n␈ε#),␈↓ ¬␈ε#.␈αε.␈α¬.␈↓ ¬:␈ε#be␈αλan␈αλi␈α↓n␈α␈|n␈α␈ite␈α	seq␈α␈uen␈α␈ce␈αλof␈α	sta␈α␈temen␈α}ts␈α	ab␈α␈ou␈α␈t␈α	m␈α␈u␈α␈tua␈α␈l␈α↓ly
␈β⊂&␈↓ βa␈ε&1␈↓ ∧@␈ε&2
␈β⊂B␈↓ ↓H␈ε#d␈α␈isjoin␈α␈t␈αev␈α␈en␈α}ts;␈α∞i.e.,␈↓ βr␈ε)S␈↓ ∧∂␈ε#(␈ε)n␈ε#␈α↓)␈αand␈↓ ¬	␈ε)S␈↓ ¬*␈ε#(␈ε)n␈ε#␈α↓)␈αcan␈α␈n␈α↓o␈α␈t␈α
sim␈α␈ulta␈α␈neo␈α␈usly␈αbe␈αtrue␈αif␈ε)␈α
i␈ε7␈α≤␈↓ 
∂␈ε)j␈↓ 
!␈ε#.␈α∪Assume
␈β⊂N␈↓ ∧¬␈ε,i␈↓ ¬≥␈ε,j
␈β⊂h␈↓ α5␈ε"(␈↓ β
␈ε")␈↓ πF␈ε"(␈↓ 
s␈ε")
␈β⊂j␈↓ ↓H␈ε#th␈α␈at␈↓ α⊃␈ε#P␈↓ α(␈ε#r␈↓ αA␈ε)S␈↓ αb␈ε#(␈ε)n␈ε#␈α↓)␈↓ β&␈ε#e␈α␈xists␈α
fo␈α␈r␈α
eac␈α␈h␈↓ ¬π␈ε)j␈↓ ¬$␈ε7∃␈ε#␈α1.␈α∀S␈α␈h␈α↓o␈α}w␈α
th␈α␈at␈↓ π#␈ε#P␈↓ π9␈ε#r␈↓ πR␈ε)S␈↓ πt␈ε#(␈ε)n␈ε#)␈↓ λ)␈ε#i␈α↓s␈αtru␈α␈e␈αfor␈αso␈α␈me␈↓ 
_␈ε)j␈↓ 
5␈ε7∃␈ε#␈α
1␈↓ ␈ε7∃
␈β⊂u␈↓ ↓H␈ε↓P␈↓ αU␈ε,j␈↓ πf␈ε,j
␈β⊃	␈↓ π#␈∧⊃	π#α$
␈β⊃∂␈↓ αL␈ε"(␈↓ β%␈ε")
␈β⊃⊃␈↓ α)␈ε#P␈↓ α?␈ε#r␈↓ αX␈ε)S␈↓ αz␈ε#(␈ε)n␈ε#␈α↓)␈↓ β1␈ε#,␈αan␈α␈d␈αg␈α␈i␈α↓v␈α}e␈αan␈αe␈α␈xa␈α␈mple␈αto␈αsho␈α␈w␈αth␈α␈at␈αeq␈α␈ualit␈α␈y␈αn␈α␈eed␈α
n␈α↓ot␈αhold␈α␈.
␈β⊃≥␈↓ αl␈ε,j
␈β⊃%␈↓ ↓n␈ε,j␈↓ ↓{␈ε:∃␈ε&␈α↓1
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"167
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα"␈↓ 	α␈ε"(␈↓ 	d␈ε")
␈βα$␈↓ ↓g␈ε37.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈27␈↓ αt␈ε#]␈α⊗Let␈ε7␈α
f␈↓ β`␈ε)S␈↓ ∧␈ε#(␈ε)n␈ε#␈α↓)␈ε7g␈ε#␈α
be␈α
a␈α
family␈α
of␈αs␈α␈tateme␈α␈n␈α␈ts␈α
such␈α
th␈α␈at␈↓ λ↑␈ε#P␈↓ λt␈ε#r␈↓ 	∞␈ε)S␈↓ 	9␈ε#(␈ε)n␈ε#)␈↓ 	z␈ε#ex␈α␈i␈α↓sts␈α
for␈α
all
␈βα0␈↓ βt␈ε,i␈↓ β}␈ε,j␈↓ 	!␈ε,i␈↓ 	+␈ε,j
␈βαL␈↓ ↓H␈ε)i␈ε#␈α␈,␈↓ ↓b␈ε)j␈↓ αα␈ε7∃␈ε#␈α∞1.␈α_Assu␈α␈me␈α∞t␈α␈hat␈α∞fo␈α␈r␈α∞all␈ε)␈α∞n␈ε#␈α∂>␈α∞0,␈↓ ε
␈ε)S␈↓ ε8␈ε#(␈ε)n␈ε#␈α↓)␈α∞is␈α∞tru␈α␈e␈α∞for␈α∞ex␈α␈actly␈α
one␈α
pair␈α∞o␈α␈f␈α∂in␈α}teger␈α␈s
␈βαW␈↓ α0␈ε↓P␈↓ ε!␈ε,i␈↓ ε+␈ε,j
␈βαq␈↓ βG␈ε"(␈↓ ∧)␈ε")␈↓ π\␈ε"(␈↓ ⊂␈ε")
␈βαs␈↓ ↓H␈ε)i␈ε#␈α␈,␈↓ ↓b␈ε)j␈↓ ↓t␈ε#.␈α⊃If␈↓ β#␈ε#P␈↓ β:␈ε#r␈↓ βS␈ε)S␈↓ β}␈ε#(␈ε)n␈ε#␈α↓)␈↓ ∧?␈ε#=␈α1␈α␈,␈α
d␈α␈oes␈αit␈αfollo␈α␈w␈αth␈α␈at␈α\␈↓ π9␈ε#P␈↓ πO␈ε#r␈↓ πh␈ε)S␈↓ λ∪␈ε#(␈ε)n␈ε#␈α↓)␈↓ λI␈ε#is␈αtrue␈αfo␈α␈r␈αsome␈↓ 
8␈ε)j␈↓ 
S␈ε7∃␈ε#␈α1␈↓ ≤␈ε#"
␈βα␈␈↓ βg␈ε,i␈↓ βp␈ε,j␈↓ π|␈ε,i␈↓ λε␈ε,j
␈ββε␈↓ ¬y␈ε↓P
␈ββπ␈↓ αV␈ε,i␈ε&,␈↓ αh␈ε,j␈↓ αv␈ε:∃␈ε&1
␈ββ ␈↓ ε}␈ε"(␈↓ π`␈ε")
␈ββ"␈↓ ↓H␈ε#e␈α␈xists␈αfor␈αall␈ε)␈αi␈ε7␈α	∃␈ε#␈α
1␈α␈,␈αa␈α␈nd␈αth␈α␈at␈αit␈αequ␈α␈als␈↓ εZ␈ε#P␈↓ εq␈ε#r␈↓ π
␈ε)S␈↓ π5␈ε#(␈ε)n␈ε#␈α↓)␈↓ πl␈ε#?
␈ββ.␈↓ π≡␈ε,i␈↓ π'␈ε,j
␈ββ6␈↓ ε∨␈ε,j␈↓ ε-␈ε:∃␈ε&1
␈ββW␈↓ ↓g␈ε38.␈↓ α␈ε#[␈ε)M1␈α␈5␈↓ α\␈ε#]␈α⊗Pro␈α␈v␈α␈e␈α(1␈α␈3).
␈ββk␈↓ π	␈ε↓P
␈β∧α␈↓ 	 ␈ε&2
␈β∧λ␈↓ ↓g␈ε39.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈20␈↓ αt␈ε#]␈α⊗Pro␈α␈v␈α}e␈αLemma␈α
E␈α↓.␈α→[␈ε0Hi␈α↓n␈α}t:␈ε#␈αC␈α↓o␈α␈nsid␈α␈er␈↓ λ↔␈ε#(␈↓ λ#␈ε)y␈↓ λY␈ε7␈␈ε)␈αλ␈↓ 	∃␈ε#)␈↓ 	.␈ε#.]
␈β∧∪␈↓ λ3␈ε,j␈↓ λA␈ε,n
␈β∧≠␈↓ π/␈ε&1␈ε:∀␈↓ πW␈ε,j␈↓ πe␈ε:∀␈ε,␈α↓m
␈β∧:␈↓ ↓;␈ε↓x
␈β∧<␈↓ ↓S␈ε31␈α␈0.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈22␈↓ αt␈ε#]␈α⊗Whe␈α␈re␈αw␈α␈as␈αth␈α␈e␈αfact␈αth␈α␈at␈ε)␈αm␈ε#␈αd␈α␈i␈α↓v␈α␈ides␈↓ π→␈ε)q␈↓ π6␈ε#u␈α␈sed␈αin␈α
the␈αp␈α␈ro␈α↓o␈α␈f␈αo␈α␈f␈αT␈α↓h␈α␈eo␈α␈rem␈αC?
␈β∧m␈↓ ↓V␈ε311.␈↓ α␈ε#[␈ε)M1␈α␈0␈↓ α\␈ε#]␈α⊗Use␈αλTheo␈α␈rem␈αλC␈αλto␈αλpro␈α}v␈α␈e␈αλtha␈α␈t␈αλi␈α↓f␈ε7␈αλh␈↓ εK␈ε)U␈↓ εs␈ε7i␈ε#␈αλis␈↓ π$␈ε71␈ε#␈α↓-d␈α␈istri␈α↓b␈α␈ute␈α␈d,␈α	so␈αλis␈αλthe␈αλsu␈α␈bse␈α␈que␈α␈nce
␈β∧y␈↓ εb␈ε,n
␈β¬∃␈↓ ↓H␈ε7h␈↓ ↓S␈ε)U␈↓ αλ␈ε7i␈ε#.
␈β¬ ␈↓ ↓i␈ε&2␈ε,n
␈β¬E␈↓ ↓V␈ε312.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈20␈↓ αt␈ε#]␈α⊗S␈α␈h␈α↓o␈α}w␈αtha␈α␈t␈αa␈↓ ∧R␈ε)k␈↓ ∧d␈ε#-distribu␈α␈ted␈αsequ␈α␈enc␈α␈e␈αpa␈α␈sses␈αth␈α␈e␈α\␈↓ λS␈ε#max␈α␈i␈α↓m␈α}um-o␈α␈f␈α↓-␈↓ 
↔␈ε)k␈↓ 
5␈ε#test,"␈αin
␈β¬g␈↓ 	_␈ε,k␈↓ 	k␈ε,k
␈β¬k␈↓ ∧⊂␈ε"(␈↓ λG␈ε")
␈β¬m␈↓ ↓H␈ε#th␈α␈e␈αfollo␈α␈wing␈αs␈α␈ense:␈↓ βm␈ε#P␈↓ ∧β␈ε#r␈↓ ∧≤␈ε)u␈ε7␈α
∀␈↓ ∧c␈ε#ma␈↓ ¬⊂␈ε#x␈↓ ¬"␈ε#(␈↓ ¬.␈ε)U␈↓ ¬U␈ε#,␈↓ ¬d␈ε)U␈↓ ε3␈ε#,␈↓ εB␈ε#.␈αε.␈αε.␈↓ εo␈ε#,␈↓ ε⎇␈ε)U␈↓ πv␈ε#)␈α
<␈↓ λ5␈ε)v␈↓ λ\␈ε#=␈↓ 	π␈ε)v␈↓ 	.␈ε7␈␈↓ 	W␈ε)u␈↓ 	y␈ε#.
␈β¬y␈↓ ¬D␈ε,n␈↓ ¬z␈ε,n␈ε&+␈α↓1␈↓ π∀␈ε,n␈ε&+␈↓ π?␈ε,k␈↓ πN␈ε:␈␈ε&␈α↓1
␈βε≤␈↓ ↓;␈ε↓x
␈βε≡␈↓ ↓S␈ε31␈α␈3.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈27␈↓ αt␈ε#]␈α⊗S␈α␈h␈α↓o␈α}w␈αth␈α␈at␈αan␈ε7␈α1␈ε#-d␈α␈i␈α↓strib␈α␈uted␈↓ ε9␈ε#[␈↓ εH␈ε#0,␈αε1)␈αseq␈α␈uen␈α␈ce␈αpa␈α␈sses␈αthe␈α\␈↓ 	F␈ε#gap␈αt␈α␈est"␈αin␈αthe
␈βεE␈↓ ↓H␈ε#fo␈α␈l␈α↓lo␈α␈win␈α␈g␈α
se␈α␈nse:␈α∞If␈α
0␈ε7␈α∀␈ε)␈α␈ε#␈α<␈↓ ∧q␈ε)␈␈↓ ¬⊃␈ε7∀␈ε#␈α1␈α
a␈α␈nd␈ε)␈αp␈ε#␈α=␈↓ εi␈ε)␈␈↓ πε␈ε7␈␈ε)␈α	␈ε#␈α␈,␈α
l␈α↓e␈α␈t␈↓ λ␈ε)f␈↓ λ∨␈ε#(0)␈α=␈α0,␈α
and␈αfo␈α␈r␈ε)␈α
n␈ε7␈α∃␈ε#␈α1␈αl␈α↓e␈α␈t
␈βεm␈↓ ↓H␈ε)f␈↓ ↓[␈ε#(␈ε)n␈ε#␈α↓)␈αbe␈α
the␈αsm␈α␈all␈α↓e␈α␈st␈αi␈α↓n␈α}tege␈α␈r␈ε)␈αm␈ε#␈αλ>␈↓ ¬6␈ε)f␈↓ ¬J␈ε#(␈ε)n␈ε7␈απ␈␈ε#␈αλ1)␈αsu␈α␈ch␈αth␈α␈at␈ε)␈α␈ε7␈α	∀␈↓ λ≤␈ε)U␈↓ λS␈ε#<␈↓ λ}␈ε)␈␈↓ 	∩␈ε#;␈αth␈α␈en
␈βεy␈↓ λ2␈ε,m
␈βπ'␈↓ λ.␈ε,k␈↓ λ=␈ε:␈␈ε&1
␈βπ-␈↓ ∧)␈ε"(␈↓ εi␈ε")
␈βπ/␈↓ ∧ε␈ε#P␈↓ ∧≤␈ε#r␈↓ ∧5␈ε)f␈↓ ∧I␈ε#(␈ε)n␈ε#)␈ε7␈αλ␈␈↓ ¬#␈ε)f␈↓ ¬7␈ε#(␈ε)n␈ε7␈αλ␈␈ε#␈αλ1␈α␈)␈α
=␈↓ εV␈ε)k␈↓ π∧␈ε#=␈ε)␈α∂p␈ε#␈α␈(␈α↓1␈ε7␈απ␈␈ε)␈απp␈↓ λ#␈ε#)␈↓ λe␈ε#.
␈βπy␈↓ ↓V␈ε314.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈25␈↓ αt␈ε#]␈α⊗S␈α␈h␈α↓o␈α}w␈α
th␈α␈at␈α	an␈ε7␈α	1␈ε#-distribu␈α␈ted␈α	se␈α␈que␈α␈nce␈α	p␈α␈asses␈α	the␈α	\␈↓ λc␈ε#run␈α	test␈α␈"␈α
in␈α	th␈α␈e␈α	foll␈α↓o␈α}w-
␈βλ!␈↓ ↓H␈ε#in␈α␈g␈α
sen␈α␈se:␈αIf␈↓ α|␈ε)f␈↓ β∂␈ε#(␈α↓0␈α␈)␈α
=␈α	1␈α
a␈α␈nd␈α	if␈α↓,␈α
for␈ε)␈α	n␈ε7␈α
∃␈ε#␈α
1␈α␈,␈↓ ε␈ε)f␈↓ ε≡␈ε#(␈ε)n␈ε#␈α↓)␈α
is␈α
th␈α␈e␈α
sma␈α␈l␈α↓lest␈α	i␈α↓n␈α}tege␈α␈r␈ε)␈α
m␈ε#␈α	>␈↓ 	h␈ε)f␈↓ 	{␈ε#(␈ε)n␈ε7␈αε␈␈ε#␈α¬1)␈α
su␈α␈ch
␈βλH␈↓ ↓H␈ε#th␈α␈at␈↓ α⊂␈ε)U␈↓ αp␈ε#>␈↓ β~␈ε)U␈↓ βI␈ε#,␈αthe␈α␈n
␈βλT␈↓ α&␈ε,m␈ε:␈␈ε&␈α↓1␈↓ β1␈ε,m
␈β	λ␈↓ β_␈ε"(␈↓ ¬W␈ε")
␈β	
␈↓ αt␈ε#P␈↓ β␈ε#r␈↓ β$␈ε)f␈↓ β7␈ε#(␈ε)␈α↓n␈ε#)␈ε7␈απ␈␈↓ ∧∩␈ε)f␈↓ ∧&␈ε#(␈ε)n␈ε7␈αλ␈␈ε#␈απ1)␈α	=␈↓ ¬E␈ε)k␈↓ ¬r␈ε#=␈α∂2␈↓ ε3␈ε)k␈↓ εE␈ε#/(␈↓ εa␈ε)k␈↓ ε{␈ε#+␈αλ1)!␈ε7␈απ␈␈ε#␈αλ2(␈↓ λ∃␈ε)k␈↓ λ/␈ε#+␈απ1)/(␈↓ 	∂␈ε)k␈↓ 	)␈ε#+␈απ2)!.
␈β	S␈↓ ↓;␈ε↓x
␈β	U␈↓ ↓S␈ε31␈α␈5.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈30␈↓ αt␈ε#]␈α⊗S␈α␈h␈α↓o␈α}w␈α	th␈α␈at␈αλan␈ε7␈απ1␈ε#-␈α↓d␈α␈istribu␈α␈ted␈αλseq␈α␈uen␈α␈ce␈αλp␈α␈asses␈αλth␈α␈e␈αλ\␈↓ λ[␈ε#co␈α␈up␈α␈on-c␈α␈oll␈α↓e␈α␈ctor's␈αλtest"
␈β	|␈↓ ↓H␈ε#wh␈α␈en␈α	ther␈α␈e␈α
are␈α	on␈α␈l␈α↓y␈α	t␈α␈w␈α␈o␈α	kind␈α␈s␈α
of␈α	cou␈α␈pon␈α␈s,␈α
i␈α↓n␈α	th␈α␈e␈α
follo␈α␈wing␈α	sen␈α␈se:␈αLet␈↓ 	4␈ε)X␈↓ 	↑␈ε#,␈↓ 	q␈ε)X␈↓ 
~␈ε#,␈↓ 
.␈ε#.␈α¬.␈αε.␈↓ 
↑␈ε#be␈α	an
␈β
λ␈↓ 	P␈ε&1␈↓ 

␈ε&2
␈β
$␈↓ ↓H␈ε71␈ε#-distrib␈α␈uted␈αbin␈α␈ary␈αse␈α␈que␈α␈nce␈α␈.␈α∩Let␈↓ ¬←␈ε)f␈↓ ¬s␈ε#(0)␈α=␈α0␈α␈,␈α
a␈α␈nd␈αf␈α↓o␈α␈r␈ε)␈αn␈ε7␈α∃␈ε#␈α1␈αlet␈↓ 	ε␈ε)f␈↓ 	~␈ε#(␈ε)n␈ε#)␈αbe␈αth␈α␈e␈αsma␈α␈ll␈α↓es␈α␈t
␈β
K␈↓ ↓H␈ε#in␈α}teger␈ε)␈α
m␈ε#␈α∞>␈↓ β∩␈ε)f␈↓ β&␈ε#(␈ε)n␈ε7␈α
␈␈ε#␈α	1)␈α∞su␈α␈ch␈α
tha␈α␈t␈ε7␈α∞f␈↓ ¬M␈ε)X␈↓ εl␈ε#,␈↓ ε{␈ε#.␈αε.␈α¬.␈↓ π'␈ε#,␈↓ π6␈ε)X␈↓ πi␈ε7g␈ε#␈α∞is␈α∞th␈α␈e␈α∞se␈α␈t␈ε7␈α∞f␈ε#0␈α␈,␈αε1␈ε7g␈ε#␈α␈.␈α↔P␈α↓r␈α␈o␈α␈v␈α␈e␈α
tha␈α␈t
␈β
W␈↓ πR␈ε,m
␈β
X␈↓ ¬h␈ε,f␈↓ ¬x␈ε&(␈ε,␈α␈n␈ε:␈␈ε&1)+1
␈β
m␈↓ ¬ε␈ε&1␈ε:␈␈↓ ¬.␈ε,k
␈β
q␈↓ ↓k␈ε"(␈↓ ∧+␈ε")
␈β
s␈↓ ↓H␈ε#P␈↓ ↓↑␈ε#r␈↓ ↓w␈ε)f␈↓ α␈ε#(␈ε)n␈ε#)␈ε7␈αλ␈␈↓ αe␈ε)f␈↓ αy␈ε#(␈ε)n␈ε7␈αλ␈␈ε#␈αλ1␈α␈)␈α
=␈↓ ∧_␈ε)k␈↓ ∧F␈ε#=␈↓ ∧v␈ε#2␈↓ ¬=␈ε#,␈αfo␈α␈r␈↓ ε¬␈ε)k␈↓ ε!␈ε7∃␈ε#␈α
2␈α␈.␈α~(Cf.␈αexer␈α␈ci␈α↓s␈α␈e␈α7.)
␈β$␈↓ ↓V␈ε316.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈38␈↓ αt␈ε#]␈α⊗Do␈α␈es␈αthe␈αco␈α␈up␈α␈on␈α␈-collector's␈αtest␈αhold␈αfor␈ε7␈α1␈ε#-distribu␈α␈ted␈αse␈α␈que␈α␈nce␈α␈s␈α
wh␈α␈en
␈βK␈↓ ↓H␈ε#th␈α␈ere␈αa␈α␈re␈αm␈α↓ore␈α
than␈α
t␈α␈w␈α␈o␈αkin␈α␈ds␈αo␈α␈f␈αc␈α␈oup␈α␈on␈α␈s?␈α→(C␈α↓f.␈αth␈α␈e␈αprev␈α␈ious␈αe␈α␈xerc␈α␈i␈α↓se␈α␈.␈α↓)
␈β|␈↓ ↓V␈ε317.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈50␈↓ αt␈ε#]␈α⊗If␈↓ β4␈ε)r␈↓ βN␈ε#is␈αa␈α␈n␈α␈y␈α
giv␈α␈en␈α
ration␈α␈al␈αn␈α}um␈α␈b␈α␈er,␈↓ π↓␈ε#F␈α⎇rank␈α␈li␈α↓n␈α
h␈α␈as␈αp␈α␈ro␈α␈v␈α␈e␈α␈d␈αth␈α␈at␈αth␈α␈e␈αse␈α␈que␈α␈nce
␈β≡␈↓ ↓b␈ε,n
␈β$␈↓ ↓H␈ε7h␈↓ ↓S␈ε)r␈↓ ↓x␈ε#mod␈↓ α=␈ε#1␈ε7␈α␈i␈ε#␈α
is␈α
not␈α
2␈α␈-distribu␈α␈ted␈α␈.␈α∀But␈αi␈α↓s␈αthe␈α␈re␈α
a␈α␈n␈α␈y␈αration␈α␈al␈α
n␈α}um␈α␈b␈α␈er␈↓ 	⊂␈ε)r␈↓ 	,␈ε#for␈α
wh␈α␈ich␈αthis␈α
se␈α␈-
␈βI␈↓ ∞␈ε&3
␈βK␈↓ ↓H␈ε#q␈α␈uen␈α␈ce␈αis␈αequ␈α␈idistribu␈α␈ted␈α␈?␈α∩In␈αp␈α␈articu␈α␈l␈α↓a␈α␈r,␈α
is␈αthe␈αse␈α␈que␈α␈nce␈αe␈α␈qu␈α␈i␈α↓d␈α␈istri␈α↓b␈α␈ute␈α␈d␈αwh␈α␈en␈↓ 
D␈ε)r␈↓ 
↑␈ε#=␈↓ ∨␈ε#?
␈β\␈↓ ∞␈ε&2
␈β↑␈↓ ∞␈∧↑∞α∞
␈βs␈↓ ↓H␈ε#[Cf.␈αK.␈↓ α7␈ε#M␈α␈ah␈α␈l␈α↓e␈α␈r,␈ε0␈αM␈α␈ath␈α␈ematik␈α⎇a␈ε3␈α
4␈ε#␈α(195␈α␈7),␈α122␈α␈↑12␈α␈4.]
␈β
"␈↓ ↓;␈ε↓x
␈β
$␈↓ ↓S␈ε31␈α␈8.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈22␈↓ αt␈ε#]␈α⊗Pro␈α␈v␈α}e␈αtha␈α␈t␈αi␈α↓f␈↓ ∧[␈ε)U␈↓ ∧␈␈ε#,␈↓ ¬∃␈ε)U␈↓ ¬:␈ε#,␈↓ ¬O␈ε#.␈αε.␈αε.␈↓ εα␈ε#i␈α↓s␈↓ ε%␈ε)k␈↓ ε7␈ε#-distribu␈α␈ted␈α␈,␈α
so␈αis␈αthe␈αse␈α␈que␈α␈nce␈↓ 
λ␈ε)V␈↓ 
)␈ε#,␈↓ 
?␈ε)V␈↓ 
a␈ε#,␈↓ 
w␈ε#.␈αε.␈α¬.␈↓ #␈ε#,
␈β
/␈↓ ∧r␈ε&0␈↓ ¬,␈ε&1␈↓ 
≤␈ε&0␈↓ 
S␈ε&1
␈β
K␈↓ ↓H␈ε#wh␈α␈ere␈↓ α(␈ε)V␈↓ αV␈ε#=␈ε7␈α
b␈ε)n␈↓ β"␈ε)U␈↓ βI␈ε7c␈ε#/␈ε)n␈ε#.
␈β
W␈↓ α<␈ε,n␈↓ β9␈ε,n
␈β
|␈↓ ↓V␈ε319.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈35␈↓ αt␈ε#]␈α⊗Co␈α␈nside␈α␈r␈αa␈αmod␈α␈i␈α↓|␈α␈catio␈α␈n␈αof␈αDe|n␈α␈ition␈αR4␈αth␈α␈at␈αreq␈α␈uires␈αthe␈αsu␈α␈bse␈α␈que␈α␈nce␈α␈s
␈β∞#␈↓ ↓H␈ε#to␈α∞b␈α␈e␈α∂o␈α␈nly␈↓ αp␈ε#1-d␈α␈i␈α↓s␈α␈tri␈α↓b␈α␈ute␈α␈d␈↓ ∧9␈ε#i␈α↓n␈α␈stea␈α␈d␈α∞of␈↓ ¬Z␈ε71␈ε#-distrib␈α␈uted␈↓ π%␈ε#.␈α→I␈α↓s␈α∞th␈α␈ere␈α∞a␈α∂se␈α␈que␈α␈nce␈α∞th␈α␈at␈α∞satis|e␈α␈s
␈β∞K␈↓ ↓H␈ε#th␈α␈is␈αw␈α␈eak␈α}er␈αd␈α␈e|␈α␈nition,␈αb␈α␈ut␈αthat␈αis␈αn␈α↓o␈α␈t␈↓ ¬|␈ε71␈ε#-distribu␈α␈ted␈↓ πG␈ε#?␈α≠(Is␈αth␈α␈e␈αw␈α␈ea␈α␈k␈α␈er␈αde|␈α␈nition␈αrea␈α␈l␈α↓ly
␈β∞s␈↓ ↓H␈ε#w␈α␈ea␈α␈k␈α␈er?␈α␈)
␈β∂≡␈↓ ¬N␈ε,n
␈β∂#␈↓ ↓V␈ε320.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈47␈↓ αt␈ε#]␈α⊗Do␈α␈es␈α⊂t␈α␈he␈α∂sequ␈α␈en␈α␈ce␈ε7␈α⊂h␈↓ ¬?␈ε)∩␈↓ ¬e␈ε#mo␈α␈d␈↓ ε)␈ε#1␈ε7i␈ε#␈α∂satisfy␈α∂De|n␈α␈iti␈α↓o␈α␈n␈α∂R4␈α∂for␈α∂alm␈α↓ost␈α∂all␈α⊂rea␈α␈l
␈β∂K␈↓ ↓H␈ε#n␈α}um␈α␈b␈α␈ers␈↓ αM␈ε)∩␈↓ αf␈ε#>␈α	1?
␈β∂|␈↓ ↓V␈ε321.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈20␈↓ αt␈ε#]␈α⊗Let␈↓ βP␈ε)S␈↓ βq␈ε#be␈αa␈α
set␈αa␈α␈nd␈α
let␈ε7␈αM␈ε#␈αbe␈α
a␈αco␈α␈l␈α↓lect␈α␈i␈α↓o␈α␈n␈αo␈α␈f␈αsub␈α␈sets␈αo␈α␈f␈↓ 	"␈ε)S␈↓ 	:␈ε#.␈α∞Su␈α␈pp␈α␈ose␈αth␈α␈at␈ε)␈αp
␈β⊂#␈↓ ↓H␈ε#is␈αa␈α
rea␈α␈l␈α↓-v␈α|alued␈α
fu␈α␈nction␈α
o␈α␈f␈αthe␈α
sets␈αin␈ε7␈α
M␈ε#,␈αsuch␈α
th␈α␈at␈ε)␈αp␈ε#␈α␈(␈↓ πY␈ε)M␈↓ π}␈ε#)␈αd␈α␈en␈α↓o␈α␈tes␈αth␈α␈e␈αp␈α␈rob␈α␈ab␈α␈i␈α↓lit␈α␈y␈α
tha␈α␈t
␈β⊂K␈↓ ↓H␈ε#a␈α	\ra␈α␈nd␈α␈omly"␈α	chosen␈α	eleme␈α␈n␈α␈t␈α
o␈α␈f␈↓ ¬!␈ε)S␈↓ ¬B␈ε#li␈α↓e␈α␈s␈α
in␈↓ ε ␈ε)M␈↓ εE␈ε#.␈α∂Gen␈α␈era␈α␈l␈α↓ize␈α	De|n␈α␈iti␈α↓o␈α␈ns␈α	B␈α
and␈α	D␈α
t␈α␈o␈α
ob␈α␈tain
␈β⊂r␈↓ ↓H␈ε#a␈α
g␈α␈o␈α↓od␈α
d␈α␈e|␈α␈nition␈α
of␈α
the␈α
co␈α␈nce␈α␈pt␈α
of␈αa␈↓ ¬`␈ε)k␈↓ ¬r␈ε#-distribu␈α␈ted␈α
seq␈α␈uen␈α␈ce␈ε7␈α
h␈↓ λ:␈ε)Z␈↓ λa␈ε7i␈ε#␈α
of␈αe␈α␈l␈α↓e␈α␈men␈α␈t␈α␈s␈αof␈↓ 
J␈ε)S␈↓ 
l␈ε#with
␈β⊂}␈↓ λP␈ε,n
␈β⊃~␈↓ ↓H␈ε#re␈α␈spec␈α␈t␈αt␈α␈o␈αthe␈αp␈α␈rob␈α␈ab␈α␈i␈α↓lit␈α␈y␈αd␈α␈i␈α↓s␈α␈tri␈α↓b␈α␈utio␈α␈n␈ε)␈αp␈ε#.
␈β∪(

␈β↓U␈↓ ↓H␈ε"168␈↓ 
}␈ε"3.5
␈β↓Z␈↓ α=␈ε.R␈α␈AND␈α␈OM␈αN␈α␈UMBERS
␈βα#␈↓ ↓;␈ε↓x
␈βα%␈↓ ↓S␈ε32␈α␈2.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈30␈↓ αt␈ε#]␈α⊗(Herm␈α␈ann␈↓ ∧/␈ε#Wey␈α␈l␈α↓.)␈α~Sho␈α␈w␈αth␈α␈at␈αth␈α␈e␈↓ π¬␈ε#[␈↓ π∀␈ε#0␈α␈,␈αε1)␈αseq␈α␈ue␈α␈nce␈↓ λd␈ε7h␈↓ λo␈ε)U␈↓ 	↔␈ε7i␈ε#␈αis␈↓ 	P␈ε)k␈↓ 	b␈ε#-distribu␈α␈ted␈αif
␈βα1␈↓ 	ε␈ε,n
␈βαM␈↓ ↓H␈ε#a␈α␈nd␈α
only␈αif
␈βββ␈↓ ∧A␈ε↓X
␈ββ⊃␈↓ ∧∧␈ε#1
␈ββ%␈↓ ¬K␈ε"(␈↓ λ}␈ε")
␈ββ'␈↓ β5␈ε#li␈α↓m␈↓ ¬↔␈ε#e␈↓ ¬&␈ε#x␈↓ ¬8␈ε#p␈↓ ¬W␈ε#2␈↓ ¬g␈ε)→␈↓ ¬|␈ε)i␈ε#␈α␈(␈↓ ε∩␈ε)c␈↓ ε/␈ε)U␈↓ ε]␈ε#+␈↓ πε␈ε7↓␈αε↓␈αε↓␈↓ π4␈ε#+␈↓ π]␈ε)c␈↓ πz␈ε)U␈↓ λs␈ε#)␈↓ 	∪␈ε#=␈α
0
␈ββ2␈↓ ε!␈ε&1␈↓ εE␈ε,n␈↓ πk␈ε,k␈↓ λ⊃␈ε,n␈ε&+␈↓ λ<␈ε,k␈↓ λK␈ε:␈␈ε&1
␈ββ:␈↓ β|␈∧β:β|α 
␈ββ>␈↓ β|␈ε)N
␈ββH␈↓ β&␈ε,N␈↓ β>␈ε:!1
␈ββY␈↓ ∧%␈ε&0␈ε:∀␈ε,␈α↓n␈ε&␈α␈<␈↓ ∧y␈ε,N
␈β∧_␈↓ ↓H␈ε#fo␈α␈r␈αev␈α␈ery␈α
set␈αof␈αin␈α␈teg␈α␈ers␈↓ ∧)␈ε)c␈↓ ∧E␈ε#,␈↓ ∧Z␈ε)c␈↓ ∧v␈ε#,␈↓ ¬
␈ε#.␈αε.␈αε.␈↓ ¬6␈ε#,␈↓ ¬K␈ε)c␈↓ ¬s␈ε#not␈αall␈αzero.
␈β∧$␈↓ ∧7␈ε&1␈↓ ∧h␈ε&2␈↓ ¬Y␈ε,k
␈β∧J␈↓ ↓V␈ε323.␈↓ α␈ε#[␈ε)M3␈α␈4␈↓ α\␈ε#]␈α⊗Sho␈α␈w␈α
t␈α␈hat␈αa␈ε)␈αb␈ε#␈α↓-a␈α␈ry␈αsequ␈α␈enc␈α␈e␈ε7␈α
h␈↓ ε(␈ε)X␈↓ εT␈ε7i␈ε#␈α
is␈↓ π∞␈ε)k␈↓ π!␈ε#-distrib␈α␈uted␈αif␈α
a␈α␈nd␈αo␈α␈nly␈αi␈α↓f␈αall␈α
of␈αthe
␈β∧V␈↓ εC␈ε,n
␈β∧r␈↓ ↓H␈ε#se␈α␈que␈α␈nce␈α␈s␈ε7␈α
h␈↓ αk␈ε)c␈↓ βπ␈ε)X␈↓ β<␈ε#+␈↓ βf␈ε)c␈↓ ∧α␈ε)X␈↓ ∧←␈ε#+␈↓ ¬	␈ε7↓␈αε↓␈αε↓␈↓ ¬9␈ε#+␈↓ ¬c␈ε)c␈↓ ε␈ε)X␈↓ ε⎇␈ε7i␈ε#␈α
are␈ε0␈α
eq␈α␈uidistrib␈α␈uted␈ε#␈α␈,␈α∞whe␈α␈nev␈α}er␈↓ 
T␈ε)c␈↓ 
p␈ε#,␈↓ π␈ε)c␈↓ #␈ε#,
␈β∧⎇␈↓ αy␈ε&1␈↓ β"␈ε,n␈↓ βt␈ε&2␈↓ ∧≥␈ε,n␈ε&+␈α↓1␈↓ ¬q␈ε,k␈↓ ε≠␈ε,n␈ε&+␈↓ εF␈ε,k␈↓ εU␈ε:␈␈ε&␈α↓1␈↓ 
b␈ε&1␈↓ ∃␈ε&2
␈β¬→␈↓ ↓H␈ε#.␈α¬.␈αε.␈↓ ↓t␈ε#,␈↓ αλ␈ε)c␈↓ α1␈ε#a␈α␈re␈αin␈α␈teg␈α␈ers␈αwith␈↓ ∧/␈ε#gc␈α␈d␈↓ ∧a␈ε#(␈↓ ∧l␈ε)c␈↓ ¬λ␈ε#,␈↓ ¬↔␈ε#.␈αε.␈αε.␈↓ ¬C␈ε#,␈↓ ¬R␈ε)c␈↓ ¬o␈ε#)␈α
=␈α	1.
␈β¬%␈↓ α↔␈ε,k␈↓ ∧z␈ε&1␈↓ ¬a␈ε,k
␈β¬K␈↓ ↓V␈ε324.␈↓ α␈ε#[␈ε)M2␈α␈5␈↓ α\␈ε#]␈α⊗Sho␈α␈w␈α
th␈α␈at␈α
a␈↓ ∧=␈ε#[␈↓ ∧L␈ε#0,␈αε1␈α␈)␈α
sequ␈α␈enc␈α␈e␈ε7␈α
h␈↓ ε*␈ε)U␈↓ εR␈ε7i␈ε#␈αi␈α↓s␈↓ π␈ε)k␈↓ π∨␈ε#-distrib␈α␈uted␈αif␈α
an␈α␈d␈α
o␈α␈nly␈αi␈α↓f␈α
a␈α␈l␈α↓l␈α
o␈α␈f␈α
the
␈β¬W␈↓ εA␈ε,n
␈β¬s␈↓ ↓H␈ε#se␈α␈que␈α␈nce␈α␈s␈ε7␈α
h␈↓ αg␈ε)c␈↓ ββ␈ε)U␈↓ β/␈ε#+␈↓ βT␈ε)c␈↓ βp␈ε)U␈↓ ∧D␈ε#+␈↓ ∧j␈ε7↓␈α¬↓␈αε↓␈↓ ¬∃␈ε#+␈↓ ¬:␈ε)c␈↓ ¬W␈ε)U␈↓ εP␈ε7i␈ε#␈α
a␈α␈re␈ε0␈α
e␈α␈qu␈α␈i␈α↓d␈α␈istri␈α↓b␈α␈ute␈α␈d␈ε#,␈α
wh␈α␈enev␈α}er␈↓ 
_␈ε)c␈↓ 
5␈ε#,␈↓ 
H␈ε)c␈↓ 
d␈ε#,␈↓ 
w␈ε#.␈αε.␈α¬.␈↓ #␈ε#,
␈β¬␈␈↓ αu␈ε&1␈↓ β~␈ε,n␈↓ βc␈ε&2␈↓ ∧π␈ε,n␈ε&+1␈↓ ¬H␈ε,k␈↓ ¬n␈ε,n␈ε&+␈↓ ε→␈ε,k␈↓ ε(␈ε:␈␈ε&1␈↓ 
'␈ε&1␈↓ 
V␈ε&2
␈βε≠␈↓ ↓H␈ε)c␈↓ ↓p␈ε#a␈α␈re␈αi␈α↓n␈α}tege␈α␈rs␈αn␈α↓o␈α␈t␈αa␈α␈ll␈αze␈α␈ro.
␈βε&␈↓ ↓V␈ε,k
␈βεM␈↓ ↓V␈ε325.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈20␈↓ αt␈ε#]␈α⊗A␈α
sequ␈α␈enc␈α␈e␈αis␈α
called␈α
a␈α
\␈↓ ¬h␈ε#white␈α
sequ␈α␈en␈α␈ce"␈α
if␈αall␈↓ λ(␈ε#se␈α␈ri␈α↓a␈α␈l␈αcorr␈α␈el␈α↓a␈α␈tions␈α
are␈α
zero␈α␈,
␈βεt␈↓ ↓H␈ε#i.e.,␈α⊃if␈α⊂th␈α␈e␈α∂equ␈α␈ation␈α∂in␈α∂Coro␈α␈l␈α↓la␈α␈ry␈α∂S␈α∂i␈α↓s␈α∂tru␈α␈e␈α⊂fo␈α␈r␈ε0␈α⊂a␈α␈l␈α↓l␈↓ π8␈ε)k␈↓ π[␈ε7∃␈ε#␈α⊃1.␈α"(By␈α∂Coro␈α␈ll␈α↓a␈α␈ry␈α∂S,␈α∂an␈ε7␈α∂1␈ε#-
␈βπ≤␈↓ ↓H␈ε#d␈α␈istribu␈α␈ted␈αseq␈α␈uen␈α␈ce␈αis␈α
wh␈α␈i␈α↓t␈α␈e.␈α↓)␈α≠Sho␈α␈w␈αtha␈α␈t␈α
if␈αa␈↓ π␈ε#[␈↓ π∂␈ε#0,␈αε1␈α␈)␈α
seq␈α␈uen␈α␈ce␈αis␈α
e␈α␈qu␈α␈i␈α↓d␈α␈istri␈α↓b␈α␈ute␈α␈d,␈α
it␈αis
␈βπC␈↓ ↓H␈ε#wh␈α␈ite␈αi␈α↓f␈αa␈α␈nd␈αo␈α␈nly␈αif
␈βπz␈↓ ∧∪␈ε↓X
␈βλπ␈↓ βa␈ε#1
␈βλ≠␈↓ ¬A␈ε&1␈↓ εj␈ε&1
␈βλ≥␈↓ β≤␈ε#lim␈↓ ∧↑␈ε#(␈↓ ∧i␈ε)U␈↓ ¬∃␈ε7␈␈↓ ¬R␈ε#)␈α↓(␈↓ ¬i␈ε)U␈↓ ε>␈ε7␈␈↓ ε|␈ε#)␈α	=␈α	0,␈↓ λ"␈ε#for␈αa␈α␈l␈α↓l␈↓ 	∧␈ε)k␈↓ 	 ␈ε7∃␈ε#␈α	1.
␈βλ)␈↓ ∧␈␈ε,j␈↓ ¬␈␈ε,j␈↓ ε
␈ε&+␈↓ ε'␈ε,k
␈βλ.␈↓ ¬A␈ε&2␈↓ εj␈ε&2
␈βλ0␈↓ β←␈∧λ0β←α∃␈↓ ¬A␈∧λ0¬Aα∞␈↓ εj␈∧λ0εjα∞
␈βλ4␈↓ β←␈ε)n
␈βλ?␈↓ β⊂␈ε,n␈ε:!1
␈βλP␈↓ β|␈ε&0␈ε:∀␈↓ ∧%␈ε,j␈↓ ∧2␈ε&<␈ε,␈α↓n
␈β	≠␈↓ ↓V␈ε326.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈34␈↓ αt␈ε#]␈α⊗(J␈α␈.␈αF␈α}ra␈α␈nk␈α␈l␈α↓in␈α␈.␈α↓)␈α↔A␈αwhite␈α
sequ␈α␈en␈α␈ce,␈αas␈α
de|␈α␈ne␈α␈d␈α
i␈α↓n␈α
th␈α␈e␈αp␈α␈reviou␈α␈s␈αe␈α␈xerc␈α␈i␈α↓se␈α␈,␈αcan
␈β	C␈↓ ↓H␈ε#d␈α␈e|␈α␈nitely␈αfail␈αto␈αb␈α␈e␈αran␈α␈dom␈α␈.␈α⊂Let␈↓ ¬-␈ε)U␈↓ ¬R␈ε#,␈↓ ¬g␈ε)U␈↓ ε␈ε#,␈↓ ε ␈ε#.␈α¬.␈αε.␈↓ εR␈ε#b␈α␈e␈αa␈α␈n␈ε7␈α1␈ε#-distribu␈α␈ted␈αseq␈α␈uen␈α␈ce,␈αan␈α␈d␈αde|␈α␈ne
␈β	N␈↓ ¬D␈ε&0␈↓ ¬⎇␈ε&1
␈β	j␈↓ ↓H␈ε#th␈α␈e␈αseq␈α␈uen␈α␈ce␈↓ β
␈ε)V␈↓ β,␈ε#,␈↓ β@␈ε)V␈↓ βb␈ε#,␈↓ βv␈ε#.␈αε.␈αε.␈↓ ∧(␈ε#as␈αfollo␈α␈ws:
␈β	v␈↓ β≡␈ε&0␈↓ βT␈ε&1
␈β
1␈↓ β5␈ε#(␈↓ β@␈ε)V␈↓ ∧≠␈ε#,␈↓ ∧*␈ε)V␈↓ ∧]␈ε#)␈α	=␈α
(␈↓ ¬'␈ε)U␈↓ ε∧␈ε#,␈↓ ε∪␈ε)U␈↓ εH␈ε#)␈↓ π≠␈ε#i␈α↓f␈α(␈↓ πF␈ε)U␈↓ λ$␈ε#,␈↓ λ2␈ε)U␈↓ λh␈ε#)␈ε7␈α	2␈ε)␈α	G␈ε#␈α↓,
␈β
=␈↓ βU␈ε&2␈ε,␈α␈n␈ε:␈␈ε&␈α↓1␈↓ ∧>␈ε&2␈ε,n␈↓ ¬>␈ε&2␈ε,␈α␈n␈ε:␈␈ε&␈α↓1␈↓ ε*␈ε&2␈ε,n␈↓ π]␈ε&2␈ε,n␈ε:␈α␈␈␈ε&␈α↓1␈↓ λI␈ε&2␈ε,n
␈β
Y␈↓ β5␈ε#(␈↓ β@␈ε)V␈↓ ∧≠␈ε#,␈↓ ∧*␈ε)V␈↓ ∧]␈ε#)␈α	=␈α
(␈↓ ¬'␈ε)U␈↓ ¬\␈ε#,␈↓ ¬k␈ε)U␈↓ εH␈ε#)␈↓ π≠␈ε#i␈α↓f␈α(␈↓ πF␈ε)U␈↓ λ$␈ε#,␈↓ λ2␈ε)U␈↓ λh␈ε#)␈ε7␈α	3␈ε)␈α	G␈ε#␈α↓,
␈β
e␈↓ βU␈ε&2␈ε,␈α␈n␈ε:␈␈ε&␈α↓1␈↓ ∧>␈ε&2␈ε,n␈↓ ¬>␈ε&2␈ε,␈α␈n␈↓ εα␈ε&2␈ε,␈α␈n␈ε:␈␈ε&␈α↓1␈↓ π]␈ε&2␈ε,n␈ε:␈α␈␈␈ε&␈α↓1␈↓ λI␈ε&2␈ε,n
␈β ␈↓ ¬≤␈ε&1␈↓ π3␈ε&1
␈β"␈↓ ↓H␈ε#wh␈α␈ere␈ε)␈αG␈ε#␈αis␈αthe␈α
set␈↓ β[␈ε7f␈↓ βq␈ε#(␈ε)x␈ε#,␈↓ ∧≡␈ε)y␈↓ ∧/␈ε#)␈ε7␈α
j␈ε)␈α	x␈ε7␈απ␈␈↓ ¬7␈ε7∀␈↓ ¬b␈ε)y␈↓ ¬⎇␈ε7∀␈ε)␈α	x␈↓ εE␈ε#o␈α␈r␈↓ εm␈ε)x␈ε#␈απ+␈↓ πN␈ε7∀␈↓ πy␈ε)y␈↓ λ⊂␈ε7g␈↓ λ!␈ε#.␈α∂S␈α␈ho␈α␈w␈αtha␈α␈t␈α→(a)␈↓ 
→␈ε)V␈↓ 
;␈ε#,␈↓ 
O␈ε)V␈↓ 
q␈ε#,␈↓ ε␈ε#.␈α¬.␈αε.
␈β.␈↓ 
-␈ε&0␈↓ 
c␈ε&1
␈β3␈↓ ¬≤␈ε&2␈↓ π3␈ε&2
␈β5␈↓ ¬≤␈∧5¬≤α∞␈↓ π3␈∧5π3α∞
␈βG␈↓ πI␈ε&5
␈βJ␈↓ ↓H␈ε#is␈αe␈α␈quid␈α␈i␈α↓s␈α␈tri␈α↓b␈α␈ute␈α␈d␈αa␈α␈nd␈α
white;␈α↔(␈α↓b␈α␈)␈↓ ¬2␈ε#P␈↓ ¬H␈ε#r␈↓ ¬U␈ε#(␈↓ ¬`␈ε)V␈↓ ε∞␈ε#>␈↓ ε9␈ε)V␈↓ πε␈ε#)␈α	=␈↓ πZ␈ε#.␈α_(This␈αp␈α␈oin␈α␈ts␈α
out␈α
the␈α
w␈α␈eak␈α␈nes␈α␈s
␈βU␈↓ ¬t␈ε,n␈↓ εM␈ε,n␈ε&+1
␈βZ␈↓ πI␈ε&8
␈β]␈↓ πI␈∧]πIα∞
␈βq␈↓ ↓H␈ε#o␈α␈f␈αt␈α␈he␈αseria␈α␈l␈αco␈α␈rrelation␈α
test.)
␈β$␈↓ ↓V␈ε327.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈48␈↓ αt␈ε#]␈α⊗Wha␈α␈t␈αis␈αt␈α␈he␈αhigh␈α␈est␈αp␈α␈ossib␈α␈l␈α↓e␈αv␈α⎇alu␈α␈e␈αfo␈α␈r␈↓ πE␈ε#P␈↓ π[␈ε#r␈↓ πh␈ε#(␈↓ πs␈ε)V␈↓ λ"␈ε#>␈↓ λN␈ε)V␈↓ 	≠␈ε#)␈αin␈αan␈αeq␈α␈uid␈α␈i␈α↓strib␈α␈-
␈β/␈↓ λλ␈ε,n␈↓ λb␈ε,n␈ε&+1
␈βK␈↓ ↓H␈ε#u␈α␈ted␈α␈,␈αwh␈α␈i␈α↓te␈α	sequ␈α␈enc␈α␈e?␈α∃(D.␈↓ ∧N␈ε#Co␈α␈pp␈α␈ersmith␈α	has␈α
c␈α␈on␈α␈structe␈α␈d␈α
su␈α␈ch␈α
a␈α	sequ␈α␈enc␈α␈e␈α
ach␈α␈ieving␈α	the
␈βp␈↓ α ␈ε&7
␈βs␈↓ ↓H␈ε#v␈α|alue␈↓ α2␈ε#.)
␈β
β␈↓ α ␈ε&8
␈β
ε␈↓ α ␈∧
εα α∞
␈β
#␈↓ ↓;␈ε↓x
␈β
%␈↓ ↓S␈ε32␈α␈8.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈21␈↓ αt␈ε#]␈α⊗Use␈α	the␈α
se␈α␈que␈α␈nce␈α	(␈α↓1␈α␈1)␈α
to␈α	con␈α␈struct␈α	a␈↓ π%␈ε#[␈↓ π4␈ε#0,␈αε1␈α␈)␈α
sequ␈α␈enc␈α␈e␈α
tha␈α␈t␈α
is␈α
3-distrib␈α␈uted␈α␈,
␈β
J␈↓ βw␈ε&1␈↓ ∧K␈ε&3
␈β
L␈↓ ↓H␈ε#fo␈α␈r␈αw␈α↓h␈α␈ich␈↓ α\␈ε#P␈↓ αr␈ε#r␈↓ α␈␈ε#(␈↓ β
␈ε)U␈↓ βI␈ε7∃␈↓ ∧λ␈ε#)␈α
=␈↓ ∧]␈ε#.
␈β
X␈↓ β!␈ε&2␈ε,n
␈β
]␈↓ βw␈ε&2␈↓ ∧K␈ε&4
␈β
←␈↓ βw␈∧
←βwα∞␈↓ ∧K␈∧
←∧Kα∞
␈β
␈␈↓ ↓V␈ε329.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈34␈↓ αt␈ε#]␈α⊗Let␈↓ βP␈ε)X␈↓ βy␈ε#,␈↓ ∧
␈ε)X␈↓ ∧7␈ε#,␈↓ ∧K␈ε#.␈αε.␈αε.␈↓ ∧⎇␈ε#be␈α
a␈α(2␈↓ ¬a␈ε)k␈↓ ¬s␈ε#)␈α↓-d␈α␈istri␈α↓b␈α␈ut␈α␈ed␈αb␈α␈i␈α↓n␈α␈ary␈α
sequ␈α␈enc␈α␈e.␈α∂Sho␈α␈w␈αtha␈α␈t
␈β∞
␈↓ βk␈ε&0␈↓ ∧)␈ε&1
␈β∞H␈↓ ε`␈ε↓∩␈↓ πZ␈ε↓∪␈↓ πp␈ε↓≡
␈β∞L␈↓ ε_␈ε#1␈↓ εv␈ε#2␈↓ πε␈ε)k␈↓ π ␈ε7␈␈ε#␈αλ1
␈β∞[␈↓ λ&␈ε&2␈↓ λ4␈ε,k
␈β∞c␈↓ ∧(␈∧∞c∧(α$␈↓ ∧(␈ε#Pr␈↓ ∧K␈ε#(␈↓ ∧W␈ε)X␈↓ ¬~␈ε#=␈α	0)␈ε7␈α	∀␈↓ ε3␈ε#+␈↓ λ∃␈ε#2␈↓ λC␈ε#.
␈β∞n␈↓ ∧r␈ε&2␈ε,n
␈β∞v␈↓ ε_␈∧∞vε_α⊃
␈β∞y␈↓ ε_␈ε#2␈↓ π∨␈ε)k
␈β∂O␈↓ ↓;␈ε↓x
␈β∂Q␈↓ ↓S␈ε33␈α␈0.␈↓ α␈ε#[␈ε)M3␈α␈9␈↓ α\␈ε#]␈α⊗Con␈α␈struct␈αa␈α
bina␈α␈ry␈αseq␈α␈uen␈α␈ce␈αth␈α␈at␈αis␈α(2␈↓ π)␈ε)k␈↓ π<␈ε#)-d␈α␈i␈α↓s␈α␈tri␈α↓b␈α␈ute␈α␈d,␈αan␈α␈d␈αfor␈αwh␈α␈i␈α↓c␈α␈h
␈β⊂≠␈↓ ε`␈ε↓∩␈↓ πZ␈ε↓∪␈↓ πp␈ε↓≡
␈β⊂∨␈↓ ε_␈ε#1␈↓ εv␈ε#2␈↓ ππ␈ε)k␈↓ π ␈ε7␈␈ε#␈αλ1
␈β⊂.␈↓ λ&␈ε&2␈↓ λ4␈ε,k
␈β⊂5␈↓ ∧(␈ε#P␈↓ ∧>␈ε#r␈↓ ∧L␈ε#(␈↓ ∧W␈ε)X␈↓ ¬~␈ε#=␈α
0␈α␈)␈α
=␈↓ ε4␈ε#+␈↓ λ∃␈ε#2␈↓ λC␈ε#.
␈β⊂A␈↓ ∧r␈ε&2␈ε,n
␈β⊂H␈↓ ε_␈∧⊂Hε_α⊃
␈β⊂L␈↓ ε_␈ε#2␈↓ π∨␈ε)k
␈β⊃→␈↓ ↓H␈ε#(Th␈α␈erefore␈α
the␈αine␈α␈qu␈α␈ali␈α↓t␈α␈y␈α
i␈α↓n␈α
the␈αp␈α␈rev␈α␈i␈α↓o␈α␈us␈αex␈α␈ercise␈αis␈αthe␈α
best␈αp␈α␈ossible.)
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"169
␈β↓Z␈↓ ¬{␈ε.WHA␈α⎇T␈α
IS␈α
A␈α
RAN␈α␈DOM␈α
SEQU␈α␈ENCE?
␈βα"␈↓ ≠␈ε&1
␈βα%␈↓ ↓V␈ε331.␈↓ α␈ε#[␈ε)M3␈α␈0␈↓ α\␈ε#]␈α⊗Sho␈α␈w␈αtha␈α␈t␈↓ ∧≥␈ε#[␈↓ ∧,␈ε#0␈α␈,␈αε1)␈αsequ␈α␈en␈α␈ces␈αexist␈αtha␈α␈t␈αsatisfy␈αDe|␈α␈nition␈αR5␈α␈,␈αy␈α␈e␈α␈t␈↓ 
≤␈ε)↔␈↓ 
>␈ε#/␈ε)␈α␈n␈ε7␈α
∃
␈βα0␈↓ 
-␈ε,n
␈βα5␈↓ ≠␈ε&2
␈βα8␈↓ ≠␈∧α8≠α∞
␈βαJ␈↓ 	~␈ε&1
␈βαL␈↓ ↓H␈ε#fo␈α␈r␈α
all␈ε)␈α
n␈ε#␈α>␈α
0␈α␈,␈α∞wh␈α␈ere␈↓ ∧∧␈ε)↔␈↓ ∧2␈ε#i␈α↓s␈αthe␈αn␈α␈u␈α␈m␈α␈ber␈αof␈↓ ε3␈ε)j␈↓ εP␈ε#<␈ε)␈α
n␈ε#␈α
fo␈α␈r␈α
which␈↓ λ6␈ε)U␈↓ λi␈ε#<␈↓ 	,␈ε#.␈α≥(Th␈α␈i␈α↓s␈αmigh␈α}t␈α
be
␈βαX␈↓ ∧∃␈ε,n␈↓ λM␈ε,n
␈βα]␈↓ 	~␈ε&2
␈βα`␈↓ 	~␈∧α`	~α∞
␈βαt␈↓ ↓H␈ε#c␈α␈onsid␈α␈ered␈α
a␈αn␈α↓o␈α␈nra␈α␈nd␈α␈om␈αp␈α␈rop␈α␈ert␈α␈y␈αof␈αth␈α␈e␈αsequ␈α␈en␈α␈ce.)
␈ββ&␈↓ ↓V␈ε332.␈↓ α␈ε#[␈ε)M2␈α␈4␈↓ α\␈ε#]␈α⊗Giv␈α␈en␈α
th␈α␈at␈ε7␈α
h␈↓ ∧-␈ε)X␈↓ ∧Y␈ε7i␈ε#␈αis␈α
a␈α
\ran␈α␈do␈α␈m"␈ε)␈α
b␈ε#-ary␈α
seq␈α␈uen␈α␈ce␈α
ac␈α␈cord␈α␈i␈α↓n␈α␈g␈α
to␈α
De|␈α␈nition␈α
R5␈α␈,
␈ββ2␈↓ ∧I␈ε,n
␈ββN␈↓ ↓H␈ε#a␈α␈nd␈α∂th␈α␈at␈ε7␈α∂R␈ε#␈α∂is␈α⊂a␈α∂c␈α␈omp␈α␈uta␈α␈ble␈α∂sub␈α␈sequ␈α␈en␈α␈ce␈α∂rule␈α∂th␈α␈at␈α∂spec␈α␈i␈α↓|␈α␈es␈α∂an␈α∂in␈α␈|n␈α␈i␈α↓te␈α∂su␈α␈bse␈α␈que␈α␈nce
␈ββv␈↓ ↓H␈ε7h␈↓ ↓S␈ε)X␈↓ ↓␈␈ε7iR␈ε#,␈α
sho␈α␈w␈α
th␈α␈at␈αthe␈αlatte␈α␈r␈α
su␈α␈bseq␈α␈ue␈α␈nce␈αis␈α
not␈αon␈α␈ly␈α1-distribu␈α␈ted␈α␈,␈α
i␈α↓t␈αis␈α
\␈α␈ran␈α␈dom␈α␈"␈αby
␈β∧↓␈↓ ↓n␈ε,n
␈β∧≥␈↓ ↓H␈ε#De␈α␈|n␈α␈i␈α↓tio␈α␈n␈αR5.
␈β∧P␈↓ ↓V␈ε333.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈22␈↓ αt␈ε#]␈α⊗Let␈ε7␈αh␈↓ β[␈ε)U␈↓ ∧∞␈ε7i␈ε#␈αa␈α␈nd␈ε7␈αh␈↓ ∧q␈ε)U␈↓ ¬$␈ε7i␈ε#␈αbe␈αin|␈α␈nite␈αdisjo␈α␈i␈α↓n␈α}t␈αsu␈α␈bse␈α␈que␈α␈nce␈α␈s␈αof␈αa␈αsequ␈α␈en␈α␈ce␈ε7␈αh␈↓ 
q␈ε)U␈↓ _␈ε7i␈ε#.
␈β∧[␈↓ βr␈ε,r␈↓ ¬λ␈ε,s␈↓ π␈ε,n
␈β∧b␈↓ β}␈ε-n␈↓ ¬∀␈ε-n
␈β∧w␈↓ ↓H␈ε#(Th␈α}us,␈↓ α/␈ε)r␈↓ αT␈ε#<␈↓ α␈␈ε)r␈↓ β$␈ε#<␈↓ βO␈ε)r␈↓ βt␈ε#<␈↓ ∧∨␈ε7↓␈αε↓␈α¬↓␈↓ ∧O␈ε#an␈α␈d␈↓ ¬∞␈ε)s␈↓ ¬5␈ε#<␈↓ ¬`␈ε)s␈↓ επ␈ε#<␈↓ ε1␈ε)s␈↓ εX␈ε#<␈↓ πα␈ε7↓␈αε↓␈αε↓␈↓ π3␈ε#are␈α	inc␈α␈reasing␈α	seq␈α␈uen␈α␈ces␈α	of␈α
in␈α}teger␈α␈s
␈β¬β␈↓ α=␈ε&0␈↓ β
␈ε&1␈↓ β]␈ε&2␈↓ ¬≡␈ε&0␈↓ ¬p␈ε&1␈↓ εA␈ε&2
␈β¬∨␈↓ ↓H␈ε#a␈α␈nd␈↓ απ␈ε)r␈↓ α6␈ε7≤␈↓ αa␈ε)s␈↓ β␈ε#for␈α
a␈α␈n␈α␈y␈ε)␈α	m␈ε#,␈ε)␈αεn␈ε#.)␈α∀L␈α↓e␈α␈t␈ε7␈α
h␈↓ ¬+␈ε)U␈↓ ¬]␈ε7i␈ε#␈α
b␈α␈e␈α
th␈α␈e␈α
c␈α␈om␈α␈b␈α␈i␈α↓n␈α␈ed␈α	sub␈α␈sequ␈α␈en␈α␈ce,␈α
so␈α
th␈α␈at␈↓ 
~␈ε)t␈↓ 
=␈ε#<␈↓ 
h␈ε)t␈↓ ␈ε#<
␈β¬*␈↓ α∃␈ε,m␈↓ αq␈ε,n␈↓ ¬B␈ε,t␈↓ 
&␈ε&0␈↓ 
t␈ε&1
␈β¬1␈↓ ¬M␈ε-n
␈β¬F␈↓ ↓H␈ε)t␈↓ ↓k␈ε#<␈↓ α⊗␈ε7↓␈αε↓␈α¬↓␈↓ αE␈ε#an␈α␈d␈α	th␈α␈e␈α	se␈α␈t␈ε7␈α	f␈↓ β⎇␈ε)t␈↓ ∧~␈ε7g␈ε#␈α	=␈ε7␈α
f␈↓ ∧o␈ε)r␈↓ ¬∞␈ε7g␈αβ[␈ααf␈↓ ¬M␈ε)s␈↓ ¬n␈ε7g␈ε#␈α␈.␈α∂S␈α␈ho␈α␈w␈α	th␈α␈at␈α	if␈↓ πO␈ε#P␈↓ πe␈ε#r␈↓ πr␈ε#(␈↓ π⎇␈ε)U␈↓ λ9␈ε72␈ε)␈α
A␈ε#)␈α	=␈↓ 	0␈ε#P␈↓ 	F␈ε#r␈↓ 	S␈ε#(␈↓ 	↑␈ε)U␈↓ 
≠␈ε72␈ε)␈α	A␈ε#)␈α
=␈ε)␈α	p␈ε#,
␈β¬R␈↓ ↓T␈ε&2␈↓ ∧	␈ε,n␈↓ ∧}␈ε,n␈↓ ¬]␈ε,n␈↓ λ∀␈ε,r␈↓ 	u␈ε,s
␈β¬Y␈↓ λ ␈ε-n␈↓ 
α␈ε-n
␈β¬n␈↓ ↓H␈ε#th␈α␈en␈↓ α∀␈ε#P␈↓ α*␈ε#r␈↓ α7␈ε#(␈↓ αB␈ε)U␈↓ α⎇␈ε72␈ε)␈α	A␈ε#)␈α
=␈ε)␈α	p␈ε#.
␈β¬z␈↓ αY␈ε,t
␈βε␈↓ αc␈ε-n
␈βε≡␈↓ ↓;␈ε↓x
␈βε ␈↓ ↓S␈ε33␈α␈4.␈↓ α␈ε#[␈ε)M2␈α␈5␈↓ α\␈ε#]␈α⊗De|␈α␈ne␈α
su␈α␈bseq␈α␈uen␈α␈ce␈α
ru␈α␈l␈α↓e␈α␈s␈↓ ¬t␈ε7R␈↓ ε≠␈ε#,␈↓ ε2␈ε7R␈↓ εZ␈ε#,␈↓ εq␈ε7R␈↓ π→␈ε#,␈↓ π0␈ε#.␈αε.␈α¬.␈↓ πd␈ε#su␈α␈ch␈α
th␈α␈at␈α
Algorith␈α␈m␈α
W␈α∞ca␈α␈n␈α
be
␈βε,␈↓ ε∞␈ε&1␈↓ εL␈ε&2␈↓ π␈ε&3
␈βεH␈↓ ↓H␈ε#u␈α␈sed␈α
with␈α
th␈α␈ese␈α
rules␈α
to␈α
g␈α␈i␈α↓v␈α}e␈α
an␈α
e{␈α␈ectiv␈α␈e␈α
a␈α␈l␈α↓g␈α␈orithm␈α
to␈α
c␈α␈on␈α␈struct␈α
a␈↓ 	∧␈ε#[␈↓ 	∪␈ε#0,␈α¬1)␈α
sequ␈α␈enc␈α␈e␈αsa␈α␈tis-
␈βεp␈↓ ↓H␈ε#fy␈α␈i␈α↓n␈α␈g␈αDe|␈α␈nition␈α
R1.
␈βπ ␈↓ ↓;␈ε↓x
␈βπ"␈↓ ↓S␈ε33␈α␈5.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈35␈↓ αt␈ε#]␈α⊗(D.␈αW␈α↓.␈↓ ∧∧␈ε#Lo␈α␈v␈α␈ela␈α␈nd␈α␈.␈α↓)␈α≠Sho␈α␈w␈αtha␈α␈t␈α
if␈αa␈αbina␈α␈ry␈αseq␈α␈uen␈α␈ce␈ε7␈αh␈↓ 	_␈ε)X␈↓ 	E␈ε7i␈ε#␈αis␈α
R␈α␈5-ran␈α␈do␈α␈m,
␈βπ.␈↓ 	4␈ε,n
␈βπG␈↓ 
[␈ε&1
␈βπJ␈↓ ↓H␈ε#a␈α␈nd␈α	if␈ε7␈α
h␈↓ α1␈ε)s␈↓ αQ␈ε7i␈ε#␈α
is␈α
an␈α}y␈α
co␈α␈mpu␈α␈tab␈α␈le␈α
seq␈α␈uen␈α␈ce␈α
a␈α␈s␈α
i␈α↓n␈α	De|␈α␈nition␈α	R4,␈α
the␈α␈n␈↓ λn␈∧πJλnα$␈↓ λn␈ε#Pr␈↓ 	⊃␈ε#(␈↓ 	≤␈ε)X␈↓ 	↑␈ε#=␈α	1)␈ε7␈α	∃␈↓ 
w␈ε#a␈α␈nd
␈βπU␈↓ α@␈ε,n␈↓ 	7␈ε,s
␈βπZ␈↓ 
[␈ε&2
␈βπ\␈↓ 	D␈ε-n
␈βπ]␈↓ 
[␈∧π]
[α∞
␈βπo␈↓ β5␈ε&1
␈βπq␈↓ ↓H␈ε#Pr␈↓ ↓k␈ε#(␈↓ ↓v␈ε)X␈↓ α8␈ε#=␈α	1)␈ε7␈α	∀␈↓ βG␈ε#.
␈βπ⎇␈↓ α⊃␈ε,s
␈βλα␈↓ β5␈ε&2
␈βλβ␈↓ α≡␈ε-n
␈βλ∧␈↓ β5␈∧λ∧β5α∞
␈βλ⊃␈↓ ↓H␈∧λ⊃↓Hα$
␈βλ$␈↓ ↓V␈ε336.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈30␈↓ αt␈ε#]␈α⊗Let␈ε7␈αh␈↓ β[␈ε)X␈↓ ∧π␈ε7i␈ε#␈αb␈α␈e␈αa␈αbin␈α␈ary␈αseq␈α␈uen␈α␈ce␈αth␈α␈at␈αis␈α\␈α␈ran␈α␈dom␈α␈"␈αacco␈α␈rdin␈α␈g␈αto␈αDe|␈α␈nition
␈βλ/␈↓ βw␈ε,n
␈βλK␈↓ ↓H␈ε#R6␈α␈.␈α∂Sho␈α␈w␈αth␈α␈at␈αthe␈↓ βe␈ε#[␈↓ βt␈ε#0␈α␈,␈αε1)␈αseq␈α␈uen␈α␈ce␈ε7␈αh␈↓ ¬N␈ε)U␈↓ ¬u␈ε7i␈ε#␈αd␈α␈e|␈α␈ned␈α
i␈α↓n␈α
bin␈α␈ary␈αnota␈α␈ti␈α↓o␈α␈n␈αb␈α␈y␈αth␈α␈e␈αsche␈α␈me
␈βλW␈↓ ¬e␈ε,n
␈β	↔␈↓ ¬≤␈ε)U␈↓ ¬J␈ε#=␈α
(0␈α␈.␈↓ ε~␈ε)X␈↓ εC␈ε#)
␈β	"␈↓ ¬3␈ε&0␈↓ ε5␈ε&0␈↓ εN␈ε&2
␈β	M␈↓ ¬≤␈ε)U␈↓ ¬J␈ε#=␈α
(0␈α␈.␈↓ ε~␈ε)X␈↓ εC␈ε)X␈↓ εl␈ε#)
␈β	X␈↓ ¬3␈ε&1␈↓ ε5␈ε&1␈↓ ε↑␈ε&2␈↓ εw␈ε&2
␈β
β␈↓ ¬≤␈ε)U␈↓ ¬J␈ε#=␈α
(0␈α␈.␈↓ ε~␈ε)X␈↓ εC␈ε)X␈↓ εl␈ε)X␈↓ π⊗␈ε#)
␈β
∞␈↓ ¬3␈ε&2␈↓ ε5␈ε&3␈↓ ε↑␈ε&4␈↓ πλ␈ε&5␈↓ π!␈ε&2
␈β
9␈↓ ¬≤␈ε)U␈↓ ¬J␈ε#=␈α
(0␈α␈.␈↓ ε~␈ε)X␈↓ εC␈ε)X␈↓ εl␈ε)X␈↓ π⊗␈ε)X␈↓ π?␈ε#)
␈β
D␈↓ ¬3␈ε&3␈↓ ε5␈ε&6␈↓ ε↑␈ε&7␈↓ πλ␈ε&8␈↓ π1␈ε&9␈↓ πJ␈ε&2
␈β
o␈↓ ¬≤␈ε#.␈αε.␈αε.
␈β1␈↓ ↓H␈ε#is␈αran␈α␈do␈α␈m␈αin␈αth␈α␈e␈αsens␈α␈e␈αof␈αDe|␈α␈nition␈αR6␈α␈.
␈βc␈↓ ↓V␈ε337.␈↓ α␈ε#[␈ε)M3␈α␈7␈↓ α\␈ε#]␈α⊗(D.␈αCo␈α␈pp␈α␈ersmith.)␈α~D␈α␈e|n␈α␈e␈αa␈αsequ␈α␈en␈α␈ce␈αtha␈α␈t␈αsa␈α␈tis|es␈αDe|␈α␈nition␈α
R4␈αbu␈α␈t␈αn␈α↓o␈α␈t
␈β␈↓ ↓H␈ε#De␈α␈|n␈α␈i␈α↓tio␈α␈n␈αλR5.␈α∞[␈ε0Hi␈α↓n␈α}t:␈ε#␈α
Con␈α␈sider␈αλch␈α␈an␈α␈ging␈↓ ε⊂␈ε)U␈↓ ε4␈ε#,␈↓ εF␈ε)U␈↓ εk␈ε#,␈↓ ε⎇␈ε)U␈↓ π!␈ε#,␈↓ π3␈ε)U␈↓ πX␈ε#,␈↓ πj␈ε#.␈α¬.␈αε.␈↓ λ→␈ε#in␈αλa␈απtruly␈αλran␈α␈do␈α␈m␈αλsequ␈α␈en␈α␈ce.]
␈β⊗␈↓ ε&␈ε&0␈↓ ε]␈ε&1␈↓ π∪␈ε&4␈↓ πJ␈ε&9
␈β=␈↓ ↓V␈ε338.␈↓ α␈ε#[␈ε)M4␈α␈9␈↓ α\␈ε#]␈α⊗(A.␈α
N.␈α
Ko␈α␈l␈α↓mogo␈α␈ro␈α␈v.)␈α≤Giv␈α␈e␈α␈n␈↓ ε'␈ε)N␈↓ εG␈ε#,␈ε)␈α
n␈ε#␈α
a␈α␈nd␈ε)␈α∂␈ε#,␈α
wh␈α␈at␈αi␈α↓s␈αthe␈αsma␈α␈l␈α↓lest␈αn␈α␈u␈α␈m␈α␈b␈α␈er␈α
o␈α␈f
␈βe␈↓ ↓H␈ε#a␈α␈l␈α↓g␈α␈orith␈α␈ms␈α
i␈α↓n␈αa␈α
set␈ε3␈α
A␈ε#␈α
su␈α␈ch␈α
th␈α␈at␈α
n␈α↓o␈α(␈ε)n␈ε#␈α↓,␈ε)␈αε∂␈ε#␈α␈)␈α↓-ra␈α␈nd␈α␈om␈α
b␈α␈inary␈αsequ␈α␈enc␈α␈es␈α
of␈α
leng␈α␈th␈↓ 
;␈ε)N␈↓ 
h␈ε#ex␈α␈i␈α↓s␈α␈t
␈β
␈↓ ↓H␈ε#with␈α
respe␈α␈ct␈α∞to␈ε3␈α
A␈ε#␈α↓?␈α≡(If␈α∞exa␈α␈ct␈α∞form␈α}ulas␈α∞ca␈α␈nnot␈α∞b␈α␈e␈α∞g␈α␈i␈α↓v␈α}en,␈α∂c␈α␈an␈α
asym␈α␈pto␈α␈ti␈α↓c␈α
form␈α␈u␈α␈l␈α↓a␈α␈s␈α∞be
␈β
4␈↓ ↓H␈ε#fo␈α␈un␈α␈d?␈α_Th␈α␈e␈α∞p␈α␈oin␈α␈t␈α∞of␈α∞th␈α␈is␈α∞prob␈α␈lem␈α∞is␈α∞to␈α∞d␈α␈i␈α↓sc␈α␈o␈α␈v␈α␈e␈α␈r␈α∞h␈α↓o␈α␈w␈α∞clo␈α␈se␈α∞the␈α∞b␈α␈ou␈α␈nd␈α
(35)␈α∞co␈α␈mes␈α∞to
␈β
[␈↓ ↓H␈ε#b␈α␈eing␈α
\be␈α␈st␈αposs␈α␈i␈α↓b␈α␈le.␈α↓"␈α␈)
␈β∞∞␈↓ ↓V␈ε339.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈45␈↓ αt␈ε#]␈α⊗(W.␈α⊂M.␈↓ ∧∞␈ε#Sch␈α␈midt.)␈α"Let␈↓ ¬}␈ε)U␈↓ ε5␈ε#be␈α∂a␈↓ ππ␈ε#[␈↓ π⊗␈ε#0␈α␈,␈αε1)␈α∂sequ␈α␈enc␈α␈e,␈α⊃an␈α␈d␈α∂l␈α↓e␈α␈t␈↓ 	r␈ε)↔␈↓ 
∀␈ε#(␈ε)u␈ε#␈α↓)␈α∂be␈α∂the
␈β∞→␈↓ ε∃␈ε,n␈↓ 
β␈ε,n
␈β∞5␈↓ ↓H␈ε#n␈α}um␈α␈b␈α␈er␈αof␈α
non␈α␈neg␈α␈ativ␈α␈e␈αin␈α␈teg␈α␈ers␈↓ ¬"␈ε)j␈↓ ¬?␈ε7∀␈ε)␈αn␈ε#␈α
su␈α␈ch␈αth␈α␈at␈α
0␈ε7␈α∀␈↓ πm␈ε)U␈↓ λ≥␈ε#<␈ε)␈αu␈ε#.␈α∀Pro␈α␈v␈α}e␈α
th␈α␈at␈αthere␈αis␈α
a
␈β∞A␈↓ λ∧␈ε,j
␈β∞]␈↓ ↓H␈ε#p␈α␈ositiv␈α␈e␈αco␈α␈nsta␈α␈n␈α␈t␈ε)␈αc␈ε#␈αsuch␈α
tha␈α␈t,␈αfo␈α␈r␈αan␈α␈y␈↓ ¬q␈ε)N␈↓ ε≤␈ε#and␈α
for␈αan␈α}y␈↓ πP␈ε#[␈↓ π←␈ε#0,␈α¬1)␈αsequ␈α␈en␈α␈ce␈ε7␈αh␈↓ 	9␈ε)U␈↓ 	a␈ε7i␈ε#,␈αw␈α␈e␈αha␈α}v␈α␈e
␈β∞i␈↓ 	P␈ε,n
␈β∂,␈↓ ¬~␈ε7j␈↓ ¬$␈ε)↔␈↓ ¬F␈ε#(␈ε)u␈ε#)␈ε7␈αλ␈␈ε)␈αλun␈ε7j␈ε#␈α
>␈ε)␈α	c␈↓ π_␈ε#l␈α↓n␈↓ π:␈ε)N
␈β∂8␈↓ ¬5␈ε,n
␈β∂y␈↓ π/␈ε"(
␈β∂{␈↓ ↓H␈ε#fo␈α␈r␈α
so␈α␈me␈ε)␈αn␈ε#␈α
an␈α␈d␈ε)␈αu␈ε#␈α
wi␈α↓t␈α␈h␈α0␈ε7␈α∀␈ε)␈αn␈ε#␈α<␈↓ ¬7␈ε)N␈↓ ¬W␈ε#,␈α
0␈ε7␈α∀␈ε)␈αu␈ε#␈α<␈α1.␈↓ π;␈ε#In␈αo␈α␈ther␈αw␈α␈ord␈α␈s,␈α
n␈α↓o␈↓ 	X␈ε#[␈↓ 	g␈ε#0,␈α¬1)␈α
se␈α␈que␈α␈nce
␈β⊂!␈↓ ∧O␈ε")
␈β⊂#␈↓ ↓H␈ε#c␈α␈an␈αb␈α␈e␈ε0␈αto␈α↓o␈ε#␈α
equ␈α␈i␈α↓d␈α␈istribu␈α␈ted.
␈β⊂S␈↓ ↓;␈ε↓x
␈β⊂U␈↓ ↓S␈ε34␈α␈0.␈↓ α␈ε#[␈ε)16␈↓ α;␈ε#]␈α⊗(I.␈αJ.␈↓ β)␈ε#Go␈α↓o␈α␈d.)␈α→C␈α↓a␈α␈n␈αa␈αv␈α|alid␈αtab␈α␈le␈αof␈αran␈α␈do␈α␈m␈αdigits␈αco␈α␈n␈α␈tain␈α
just␈αon␈α␈e␈αmisp␈α␈ri␈α↓n␈α}t?
␈β∪(/FONT#1=cmathx[XGP,SYS]=↓α∩∪~≡01@APSTXZ[\qxx/FONT#34=cmr10[XGP,SYS]=∪!"$'()*+,-./0123456789:;<=>?ABCDEFGHIJKLMNOPQRSTUVWYZ[\]↑←abcdefghijklmnopqrstuvwxyz{|⎇}}/FONT#35=cmr9[XGP,SYS]=!"'()+,-./012345679:;<=>?ABCDEFGHIJKLMNPRSTUW[\]↑abcdefghijklmnopqrstuvwxyz{||/FONT#36=cmr8[XGP,SYS]=*,.0169Aacdefghilmnoprstuwyy/FONT#37=cmr7[XGP,SYS]=()+./0123456789<</FONT#38=cmr6[XGP,SYS]=()+,0123456789<gll/FONT#39=cmr5[XGP,SYS]=01233/FONT#40=cmi10[XGP,SYS]=∂∩∀∃⊗↔→ABCINPSTUVWXYabcdefjklmnpqrstuvxyy/FONT#41=cmi9[XGP,SYS]=∂∩↔→0123456789AGHMNSTUVXZbcfijkmnpqrstuvxyy/FONT#43=cmi7[XGP,SYS]=∂Nabfjkmnpqrstxx/FONT#44=cmi6[XGP,SYS]=Nfijkmnrstt/FONT#45=cmi5[XGP,SYS]=Nkmnprtt/FONT#46=cmsc10[XGP,SYS]=?ABCDEHIMNOQRSTUWW/FONT#47=cms10[XGP,SYS]=∞⊂∩∪_"$'(),-.2456:;?ABCDEFGHIJKLMNPRSTUWZ\abcdefghijklmnopqrstuvwxyz{|}}/FONT#48=cms9[XGP,SYS]=:HMabdehiklmnoqrstuu/FONT#50=cmb10[XGP,SYS]=,-.0123456789?ABCDEFILMPQRSTWabcdefghiklmnopqrstuxy||/FONT#51=cmb9[XGP,SYS]=.0123456789AA/FONT#54=cmsy10[XGP,SYS]=↓ε∩∀∃→≤ !12RS[bcfghijp⎇⎇/FONT#55=cmsy9[XGP,SYS]=↓∀∃≤123MR[bcfghijj/FONT#57=cmsy7[XGP,SYS]=↓∀∃!011/FONT#58=cmsy6[XGP,SYS]=∀∃!1bcc/FONT#59=cmsy5[XGP,SYS]=/FONT#61=cmssb[XGP,SYS]=.35?ACDEHIMNOQRSTUWXX