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

perm filename V235.XGP[TEX,DEK] blob sn#521446 filedate 1980-07-10 generic text, type T, neo UTF8
/NOWRAPAROUND/LMAR=50/TMAR=50/RMAR=1700/BMAR=1/PMAR=0/XLINE=0/FONT#0=NGR13/USETI=000000276*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↑W␈α⎇esley␈αPublishing␈αCompan␈α␈y,␈αInc.
␈β⊃H␈↓ ε2␈ε$0
␈β∪(

␈β↓U␈↓ ↓H␈ε"142␈↓ 
}␈ε"3.5
␈β↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈βα$␈↓ ↓H␈ε=3␈α␈.5.␈α∩WHA␈α⎇T␈α
IS␈α
A␈α
RANDOM␈αSEQUENCE?
␈βαf␈↓ ↓H␈ε2A.␈α	In␈α␈troductory␈α	remarks.␈ε"␈α∞W␈α⎇e␈↓ ¬∀␈ε"ha␈α␈v␈α␈e␈α	seen␈α	in␈α	this␈α	chapter␈α	h␈α↓o␈α␈w␈α	to␈αλgenerate␈α	sequences
␈ββ=␈↓ ¬∧␈ε6h␈↓ ¬⊂␈ε(U␈↓ ¬=␈ε6i␈ε"␈α
=␈↓ ε↓␈ε(U␈↓ ε+␈ε",␈↓ εC␈ε(U␈↓ εl␈ε",␈↓ π∧␈ε(U␈↓ π.␈ε",␈↓ πF␈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∀␈↓ ¬g␈ε(U␈↓ ε ␈ε"<␈α
1,␈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	justify
␈β∧k␈↓ ↓H␈ε"this␈α⊃termin␈α↓ology,␈α∀w␈α␈e␈α⊃claimed␈α∩that␈α∩the␈α∩n␈α␈um␈α␈bers␈α∩\beha␈α␈v␈α␈e␈α⊃as␈α∩if␈α∩they␈α∩are␈α⊃truly
␈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␈α∂in␈α⊂terms␈α⊂of␈α⊂h␈α↓o␈α␈w␈α⊂m␈α␈uch␈ε/␈α⊂probabilit␈α␈y␈ε"␈α∂is␈α⊂to␈α⊂be␈α⊂attached␈α⊂to␈α∂statemen␈α␈ts
␈βλ≠␈↓ ↓H␈ε"in␈α␈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
␈βλF␈↓ ↓H␈ε"set␈α∂up␈α⊂so␈α⊂that␈α∂abstract␈α⊂probabilities␈α⊂can␈α∂be␈α⊂computed␈α⊂readily,␈α⊂but␈α⊂n␈α↓othing␈α∂is
␈βλq␈↓ ↓H␈ε"said␈αabout␈α
what␈α
probabilit␈α␈y␈α
really␈α
signi|es,␈α
or␈αh␈α↓o␈α␈w␈α
this␈α
concept␈α
can␈α
be␈αapplied
␈β	≥␈↓ ↓H␈ε"meaningfully␈αto␈αthe␈αactual␈αw␈α␈orld.␈α⊂In␈αthe␈αbo␈α↓ok␈ε/␈αProbabilit␈α␈y,␈αStatistics,␈αand␈αT␈α⎇ruth
␈β	H␈↓ ↓H␈ε"(New␈α	Y␈α⎇ork:␈αMacmillan,␈α
1957),␈α
R.␈↓ ¬S␈ε"v␈α␈on␈α	Mises␈α
discusses␈α
this␈α	situation␈α
in␈α	detail,␈α
and
␈β	s␈↓ ↓H␈ε"presen␈α␈ts␈αthe␈αview␈αthat␈αa␈αproper␈αde|nition␈αof␈αprobabilit␈α␈y␈αdepends␈αon␈αobtaining␈αa
␈β
≡␈↓ ↓H␈ε"proper␈αde|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.␈↓ αj␈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↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈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(␈αu␈ε"␈α
<␈↓ π∪␈ε(v␈↓ π3␈ε6∀␈ε"␈α
1.␈α∃If␈↓ λ;␈ε(U␈↓ λf␈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␈↓ βε␈ε",␈↓ β≡␈ε(U␈↓ βG␈ε",␈↓ β←␈ε(U␈↓ ∧λ␈ε",␈↓ ∧∨␈ε".␈αε.␈αε.␈↓ ∧O␈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␈↓ αu␈ε%0␈↓ β6␈ε%1␈↓ βw␈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(↔␈↓ ∧[␈ε"(␈ε(n␈ε"␈α␈)␈α⊂be␈α⊂the␈α⊂n␈α␈um␈α␈ber␈α∂of␈α⊂v␈α}alues␈α⊂of␈↓ λ↑␈ε(j␈↓ λo␈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(↔␈↓ ε3␈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␈↓ ¬a␈ε(↔␈↓ ¬v␈ε"(␈ε(n␈ε"␈α␈)/␈ε(n␈ε"␈α
=␈↓ πβ␈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␈)␈α
be␈α
a␈αstatemen␈α␈t␈α
about␈αthe␈α
in␈α␈teger␈ε(␈α
n␈ε"␈αand␈α
the␈αsequence␈↓ 	r␈ε(U␈↓ 
≠␈ε",␈↓ 
2␈ε(U␈↓ 
\␈ε",␈↓ 
r␈ε".␈αε.␈αε.␈↓ "␈ε";
␈βπ]␈↓ 
␈ε%1␈↓ 
K␈ε%2
␈βπ|␈↓ ↓H␈ε"for␈α
example,␈↓ β∃␈ε(S␈↓ β.␈ε"(␈ε(n␈ε")␈α
migh␈α␈t␈α
be␈αthe␈α
statemen␈α␈t␈↓ ε[␈ε"considered␈αabo␈α␈v␈α␈e,␈α
namely␈α\␈ε(u␈ε6␈α	∀␈↓ 
Q␈ε(U␈↓ λ␈ε"<
␈βλλ␈↓ 
j␈ε+n
␈βλ'␈↓ ↓H␈ε(v␈↓ ↓[␈ε"."␈α∀W␈α⎇e␈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␈↓ ¬F␈ε(j␈↓ ¬W␈ε",␈α0␈ε6␈α
∀␈↓ ε7␈ε(j␈↓ εR␈ε"<␈ε(␈α
n␈ε"␈α␈,␈αsuch␈αthat␈↓ λK␈ε(S␈↓ λd␈ε"(␈↓ λp␈ε(j␈↓ 	↓␈ε")␈αis␈αtrue.
␈β	≥␈↓ ∧c␈ε↓␈␈↓ ¬8␈ε↓↓
␈β	<␈↓ ↓H␈ε2De|nition␈α⊂A.␈ε/␈α"W␈α⎇e␈α⊃sa␈α␈y␈↓ ∧=␈ε"P␈↓ ∧U␈ε"r␈↓ ∧q␈ε(S␈↓ ¬
␈ε"(␈ε(n␈ε")␈↓ ¬X␈ε"=␈ε(␈α⊃∃␈ε/,␈α∩if␈↓ εl␈ε"lim␈↓ πs␈ε(↔␈↓ λλ␈ε"(␈ε(n␈ε")/␈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␈αterms␈α
of␈αthis␈α
n␈α↓otation,␈αthe␈α
sequence␈↓ ε|␈ε(U␈↓ π&␈ε",␈↓ π<␈ε(U␈↓ πf␈ε",␈↓ π⎇␈ε".␈αε.␈αε.␈↓ λ3␈ε"is␈α
equidistributed␈αif␈αand
␈β
]␈↓ π∃␈ε%0␈↓ πU␈ε%1
␈β
{␈↓ ↓H␈ε"only␈α
if␈↓ α6␈ε"P␈↓ αN␈ε"r␈↓ α\␈ε"(␈ε(u␈ε6␈α
∀␈↓ β5␈ε(U␈↓ βl␈ε"<␈↓ ∧~␈ε(v␈↓ ∧-␈ε")␈α
=␈↓ ∧q␈ε(v␈↓ ¬	␈ε6␈␈ε(␈α¬u␈ε",␈αfor␈αall␈αreal␈α
n␈α␈um␈α␈bers␈ε(␈αu␈ε",␈↓ λA␈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␈↓ ↓q␈ε",␈↓ απ␈ε(U␈↓ α1␈ε",␈↓ αG␈ε".␈αε.␈αε.␈↓ α}␈ε"and␈↓ βD␈ε(V␈↓ βj␈ε",␈↓ ∧↓␈ε(V␈↓ ∧'␈ε",␈↓ ∧=␈ε".␈αε.␈αε.␈↓ ∧t␈ε"are␈αequidistributed␈αsequences,␈αit␈α
is␈αn␈α↓ot␈αhard␈αto␈αsh␈α↓o␈α␈w
␈β↑␈↓ ↓`␈ε%0␈↓ α ␈ε%1␈↓ βY␈ε%0␈↓ ∧⊗␈ε%1
␈β⎇␈↓ ↓H␈ε"that␈αthe␈αsequence
␈βI␈↓ ¬n␈ε%1␈↓ εG␈ε%1␈↓ πε␈ε%1␈↓ π\␈ε%1␈↓ λ5␈ε%1␈↓ λt␈ε%1
␈βM␈↓ β∧␈ε(W␈↓ β4␈ε",␈↓ βD␈ε(W␈↓ βs␈ε",␈↓ ∧β␈ε(W␈↓ ∧3␈ε",␈↓ ∧C␈ε(W␈↓ ∧r␈ε",␈↓ ¬α␈ε".␈αε.␈αε.␈↓ ¬<␈ε"=␈↓ εα␈ε(U␈↓ ε+␈ε",␈↓ ε\␈ε"+␈↓ π~␈ε(V␈↓ π@␈ε",␈↓ πo␈ε(U␈↓ λ→␈ε",␈↓ λJ␈ε"+␈↓ 	π␈ε(V␈↓ 	.␈ε",␈↓ 	F␈ε".␈αε.␈αε.␈↓ α␈ε"(3)
␈βZ␈↓ β#␈ε%0␈↓ βc␈ε%1␈↓ ∧"␈ε%2␈↓ ∧b␈ε%3␈↓ ε~␈ε%0␈↓ π/␈ε%0␈↓ λλ␈ε%1␈↓ 	≥␈ε%1
␈β↑␈↓ ¬n␈ε%2␈↓ εG␈ε%2␈↓ πε␈ε%2␈↓ π\␈ε%2␈↓ λ5␈ε%2␈↓ λt␈ε%2
␈βb␈↓ ¬n␈∧b¬nα⊂␈↓ εG␈∧bεGα⊂␈↓ πε␈∧bπεα⊂␈↓ π\␈∧bπ\α⊂␈↓ λ5␈∧bλ5α⊂␈↓ λt␈∧bλtα⊂
␈β
→␈↓ π.␈ε%1␈↓ λ	␈ε%1
␈β
≡␈↓ ↓H␈ε"is␈α∂also␈α∂equidistributed,␈α⊂since␈α⊂the␈α∂subsequence␈↓ πA␈ε(U␈↓ πk␈ε",␈↓ λ≤␈ε(U␈↓ λF␈ε",␈↓ λ`␈ε".␈αε.␈αε.␈↓ 	→␈ε"is␈α∂equidistributed
␈β
*␈↓ πZ␈ε%0␈↓ λ5␈ε%1
␈β
/␈↓ π.␈ε%2␈↓ λ	␈ε%2
␈β
2␈↓ π.␈∧
2π.α⊂␈↓ λ	␈∧
2λ	α⊂
␈β
D␈↓ βA␈ε%1␈↓ π∪␈ε%1␈↓ πa␈ε%1␈↓ λ8␈ε%1␈↓ 	ε␈ε%1
␈β
I␈↓ ↓H␈ε"bet␈α␈w␈α␈een␈α∞0␈α∞and␈↓ βT␈ε",␈α∂while␈α∞the␈α∂alternate␈α∞terms␈↓ π/␈ε"+␈↓ πt␈ε(V␈↓ λ≠␈ε",␈↓ λU␈ε"+␈↓ 	→␈ε(V␈↓ 	@␈ε",␈↓ 	Y␈ε".␈αε.␈αε.␈↓ 
	␈ε",␈α∂are␈α∞equi-
␈β
U␈↓ λ
␈ε%0␈↓ 	/␈ε%1
␈β
Z␈↓ βA␈ε%2␈↓ π∪␈ε%2␈↓ πa␈ε%2␈↓ λ8␈ε%2␈↓ 	ε␈ε%2
␈β
]␈↓ βA␈∧
]βAα⊂␈↓ π∪␈∧
]π∪α⊂␈↓ πa␈∧
]πaα⊂␈↓ λ8␈∧
]λ8α⊂␈↓ 	ε␈∧
]	εα⊂
␈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⊂W␈α⎇e␈αcan␈αrequire␈αthe␈αsequence␈αto␈αsatisfy␈αthe␈αcondition
␈β⊂H␈↓ α/␈ε"P␈↓ αG␈ε"r␈↓ αU␈ε"(␈↓ αa␈ε(u␈↓ β⊃␈ε6∀␈↓ β?␈ε(U␈↓ βv␈ε"<␈↓ ∧$␈ε(v␈↓ ∧j␈ε"and␈↓ ¬H␈ε(u␈↓ ¬w␈ε6∀␈↓ ε%␈ε(U␈↓ π	␈ε"<␈↓ π7␈ε(v␈↓ πZ␈ε")␈α
=␈α
(␈↓ λ*␈ε(v␈↓ λT␈ε6␈␈↓ 	␈ε(u␈↓ 	%␈ε")(␈↓ 	=␈ε(v␈↓ 	h␈ε6␈␈↓ 
∀␈ε(u␈↓ 
9␈ε")␈↓ α␈ε"(4)
␈β⊂T␈↓ αv␈ε%1␈↓ βW␈ε+n␈↓ ∧5␈ε%1␈↓ ¬]␈ε%2␈↓ ε>␈ε+n␈ε%+␈α␈1␈↓ πI␈ε%2␈↓ λ;␈ε%1␈↓ 	∃␈ε%1␈↓ 	O␈ε%2␈↓ 
(␈ε%2
␈β⊃_␈↓ ↓H␈ε"for␈α	an␈α␈y␈α	four␈α	n␈α␈um␈α␈bers␈↓ ∧∀␈ε(u␈↓ ∧:␈ε",␈↓ ∧N␈ε(v␈↓ ∧p␈ε",␈↓ ¬∧␈ε(u␈↓ ¬)␈ε",␈↓ ¬=␈ε(v␈↓ ¬i␈ε"with␈α	0␈ε6␈α
∀␈↓ πα␈ε(u␈↓ π1␈ε"<␈↓ π←␈ε(v␈↓ λ␈ε6∀␈ε"␈α
1,␈α
0␈ε6␈α
∀␈↓ 	*␈ε(u␈↓ 	Y␈ε"<␈↓ 
π␈ε(v␈↓ 
3␈ε6∀␈ε"␈α
1.␈α∂In
␈β⊃$␈↓ ∧)␈ε%1␈↓ ∧←␈ε%1␈↓ ¬→␈ε%2␈↓ ¬O␈ε%2␈↓ π↔␈ε%1␈↓ πq␈ε%1␈↓ 	>␈ε%2␈↓ 
→␈ε%2
␈β∪(

␈β↓U␈↓ ↓H␈ε"144␈↓ 
}␈ε"3.5
␈β↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈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␈↓ ππ␈ε(k␈↓ π≠␈ε/-distributed␈αif
␈ββv␈↓ ↓H␈ε"P␈↓ ↓`␈ε"r␈↓ ↓n␈ε"(␈↓ ↓z␈ε(u␈↓ α)␈ε6∀␈↓ αW␈ε(U␈↓ β∞␈ε"<␈↓ β<␈ε(v␈↓ β←␈ε",␈↓ βo␈ε".␈αε.␈αε.␈↓ ∧∨␈ε",␈↓ ∧/␈ε(u␈↓ ∧`␈ε6∀␈↓ ¬∞␈ε(U␈↓ ε ␈ε"<␈↓ εN␈ε(v␈↓ εq␈ε")␈α
=␈α
(␈↓ πA␈ε(v␈↓ πl␈ε6␈␈↓ λ_␈ε(u␈↓ λ=␈ε")␈↓ λO␈ε".␈αε.␈αε.␈↓ λ␈␈ε"(␈↓ 	␈ε(v␈↓ 	7␈ε6␈␈↓ 	c␈ε(u␈↓ 

␈ε")␈↓ α␈ε"(5)
␈β∧α␈↓ α∂␈ε%1␈↓ αp␈ε+n␈↓ βN␈ε%1␈↓ ∧C␈ε+k␈↓ ¬&␈ε+n␈ε%+␈↓ ¬W␈ε+k␈↓ ¬g␈ε9␈␈ε%1␈↓ ε←␈ε+k␈↓ πS␈ε%1␈↓ λ,␈ε%1␈↓ 	≥␈ε+k␈↓ 	x␈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␈↓ 
3␈ε6∀␈↓ 
a␈ε(k␈↓ 
u␈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,␈αsince␈αw␈α␈e␈αma␈α␈y␈αset␈↓ 
<␈ε(u␈↓ 
l␈ε"=␈α
0
␈β¬S␈↓ 
P␈ε+k
␈β¬r␈↓ ↓H␈ε"and␈↓ α⊂␈ε(v␈↓ αA␈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⊂W␈α⎇e
␈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␈αbet␈α␈w␈α␈een␈α
zero␈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␈α
us␈αsa␈α␈y␈αa␈αsequence␈ε6␈αh␈↓ ∧ ␈ε(X␈↓ ∧R␈ε6i␈ε"␈α
=␈↓ ¬⊗␈ε(X␈↓ ¬E␈ε",␈↓ ¬Z␈ε(X␈↓ ε	␈ε",␈↓ ε≡␈ε(X␈↓ εL␈ε",␈↓ εa␈ε".␈αε.␈αε.␈↓ π⊗␈ε"is␈αa␈α\␈↓ πh␈ε(b␈ε"-ary␈αsequence"␈αif␈αeach␈↓ 
W␈ε(X␈↓ ∀␈ε"is
␈β	→␈↓ ∧>␈ε+n␈↓ ¬4␈ε%0␈↓ ¬x␈ε%1␈↓ ε<␈ε%2␈↓ 
u␈ε+n
␈β	7␈↓ ↓H␈ε"one␈α
of␈αthe␈αin␈α␈tegers␈α0,␈α
1,␈↓ ∧?␈ε".␈αε.␈αε.␈↓ ∧o␈ε",␈ε(␈αb␈ε6␈αε␈␈ε"␈αε1.␈α∂Th␈α␈us,␈αa␈α2-ary␈α(binary)␈α
sequence␈αis␈αa␈α
sequence
␈β	c␈↓ ↓H␈ε"of␈αzeros␈αand␈αones.
␈β
∂␈↓ α␈ε"A␈↓ α3␈ε(k␈↓ αG␈ε"-digit␈α∞\␈↓ β;␈ε(b␈ε"-ary␈α
n␈α␈um␈α␈ber"␈↓ ¬+␈ε(x␈↓ ¬P␈ε(x␈↓ ¬{␈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
≤␈↓ ¬?␈ε%1␈↓ ¬d␈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∀␈↓ ∧S␈ε(k␈↓ ∧g␈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␈↓ βC␈ε"P␈↓ β[␈ε"r␈↓ βi␈ε"(␈↓ βu␈ε(X␈↓ ∧(␈ε(X␈↓ ¬
␈ε".␈αε.␈αε.␈↓ ¬=␈ε(X␈↓ εT␈ε"=␈↓ πα␈ε(x␈↓ π'␈ε(x␈↓ πQ␈ε".␈αε.␈αε.␈↓ λ↓␈ε(x␈↓ λ(␈ε")␈α
=␈α
1/␈↓ 	⊂␈ε(b␈↓ α␈ε"(6)
␈βm␈↓ ∧∪␈ε+n␈↓ ∧F␈ε+n␈ε%␈α␈+1␈↓ ¬[␈ε+n␈ε%␈α␈+␈↓ ε␈ε+k␈↓ ε≠␈ε9␈␈ε%1␈↓ π⊗␈ε%1␈↓ π;␈ε%2␈↓ λ∃␈ε+k
␈β<␈↓ ↓H␈ε/for␈αall␈ε(␈αb␈ε/-ary␈αn␈α␈um␈α␈bers␈↓ ∧≤␈ε(x␈↓ ∧A␈ε(x␈↓ ∧l␈ε".␈αε.␈αε.␈↓ ¬≤␈ε(x␈↓ ¬B␈ε/.
␈βH␈↓ ∧0␈ε%1␈↓ ∧U␈ε%2␈↓ ¬0␈ε+k
␈β
π␈↓ α␈ε"It␈α⊂is␈α∂clear␈α⊂from␈α⊂this␈α∂de|nition␈α⊂that␈α⊂if␈↓ εu␈ε(U␈↓ π∨␈ε",␈↓ π9␈ε(U␈↓ πc␈ε",␈↓ π⎇␈ε".␈αε.␈αε.␈↓ λ7␈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␈↓ ¬W␈ε6c␈ε",␈ε6␈α
b␈ε(b␈↓ ε~␈ε(U␈↓ εC␈ε6c␈ε",␈↓ εi␈ε".␈αε.␈αε.␈↓ π ␈ε"is␈α∞a␈↓ πe␈ε(k␈↓ πy␈ε"-distributed␈ε(␈α∞b␈ε"-ary␈α
sequence.
␈β
?␈↓ ↓H␈ε↓␈␈↓ ¬F␈ε%0␈↓ ε2␈ε%1
␈β
]␈↓ ↓V␈ε"If␈α⊃w␈α␈e␈α⊃set␈↓ αv␈ε(u␈↓ β-␈ε"=␈↓ βd␈ε(x␈↓ ∧π␈ε"/␈ε(b␈ε",␈↓ ∧E␈ε(v␈↓ ∧x␈ε"=␈α∪(␈↓ ¬;␈ε(x␈↓ ¬i␈ε"+␈α1)/␈ε(b␈ε",␈↓ εu␈ε(X␈↓ π9␈ε"=␈ε6␈α∪b␈ε(b␈↓ λ
␈ε(U␈↓ λ:␈ε6c␈ε",␈α∪Eq.␈α⊃(5)␈α⊃becomes␈α⊃Eq.
␈β
j␈↓ ↓|␈ε↓↓␈↓ β␈ε+j␈↓ βx␈ε+j␈↓ ∧V␈ε+j␈↓ ¬O␈ε+j␈↓ π∪␈ε+n␈↓ λ&␈ε+n
␈β∞	␈↓ ↓H␈ε"(6).␈↓ α≥␈ε"F␈α⎇urtherm␈α↓ore,␈αev␈α␈ery␈↓ ∧V␈ε(k␈↓ ∧j␈ε"-distributed␈ε(␈α
b␈ε"␈α↓-ary␈α
sequence,␈αfor␈↓ λX␈ε(k␈↓ λw␈ε">␈α
1,␈α
is␈αalso␈α
(␈↓ 
@␈ε(k␈↓ 
Y␈ε6␈␈ε"␈α¬1)-
␈β∞4␈↓ ↓H␈ε"distributed:␈α
w␈α␈e␈αλadd␈α	together␈αλthe␈α	probabilities␈α	for␈αλthe␈ε(␈α	b␈ε"-ary␈αλn␈α␈um␈α␈bers␈↓ 	↑␈ε(x␈↓ 
	␈ε".␈αε.␈αε.␈↓ 
9␈ε(x␈↓ ⊂␈ε"0,
␈β∞@␈↓ 	r␈ε%1␈↓ 
M␈ε+k␈↓ 
]␈ε9␈␈ε%1
␈β∞←␈↓ ↓H␈ε(x␈↓ ↓r␈ε".␈αε.␈αε.␈↓ α"␈ε(x␈↓ αz␈ε"1,␈↓ β"␈ε".␈αε.␈αε.␈↓ βR␈ε",␈↓ βh␈ε(x␈↓ ∧∪␈ε".␈αε.␈αε.␈↓ ∧C␈ε(x␈↓ ¬≠␈ε"(␈ε(b␈ε6␈αλ␈␈ε"␈αλ1)␈αto␈αobtain
␈β∞k␈↓ ↓\␈ε%1␈↓ α6␈ε+k␈↓ αG␈ε9␈␈ε%1␈↓ β|␈ε%1␈↓ ∧W␈ε+k␈↓ ∧h␈ε9␈␈ε%␈α␈1
␈β∂2␈↓ λX␈ε+k␈↓ λh␈ε9␈␈ε%1
␈β∂:␈↓ βS␈ε"P␈↓ βk␈ε"r␈↓ βz␈ε"(␈↓ ∧ε␈ε(X␈↓ ∧>␈ε".␈αε.␈αε.␈↓ ∧n␈ε(X␈↓ ε¬␈ε"=␈↓ ε3␈ε(x␈↓ ε↑␈ε".␈αε.␈αε.␈↓ π∞␈ε(x␈↓ πa␈ε")␈α
=␈α
1/␈↓ λI␈ε(b␈↓ 	↔␈ε".
␈β∂F␈↓ ∧$␈ε+n␈↓ ¬␈ε+n␈ε%␈α␈+␈↓ ¬<␈ε+k␈↓ ¬L␈ε9␈␈ε%2␈↓ εG␈ε%1␈↓ π"␈ε+k␈↓ π2␈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␈α
a␈αpositiv␈α␈e␈α
real␈α
n␈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␈α
regarded␈α
as␈α
a␈ε(␈α
b␈ε"-ary␈α
sequence;␈α
for␈α
example,␈↓ πv␈ε(→␈↓ λ_␈ε"corresponds␈α
to␈α
the␈α
10-ary
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"145
␈β↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈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∧#␈↓ ∂␈ε%1
␈β∧'␈↓ ↓H␈ε"is␈α
going␈αto␈α
ha␈α␈v␈α␈e␈αruns␈αof␈α
a␈αmillion␈α
consecutiv␈α␈e␈αv␈α}alues␈αeach␈α
of␈αwhich␈α
is␈αless␈αthan␈↓ "␈ε".
␈β∧8␈↓ ∂␈ε%2
␈β∧;␈↓ ∂␈∧∧;∂α⊂
␈β∧L␈↓ ε9␈ε%1000␈α↓000
␈β∧N␈↓ ε~␈ε%1
␈β∧R␈↓ ↓H␈ε"It␈α
is␈αtrue␈α
that␈α
this␈α
will␈α
happen␈α
only␈α
(␈↓ ε-␈ε")␈↓ π2␈ε"of␈α
the␈αtime,␈α∞on␈αthe␈α
a␈α␈v␈α␈erage,␈α
but
␈β∧c␈↓ ε~␈ε%2
␈β∧g␈↓ ε~␈∧∧gε~α⊂
␈β∧}␈↓ ↓H␈ε"the␈αfact␈αis␈αthat␈αit␈ε/␈αdoes␈ε"␈αhappen.␈α⊂Indeed,␈αthis␈αphen␈α↓omen␈α↓on␈αwill␈αoccur␈αin␈αan␈α␈y␈αtruly
␈β¬)␈↓ ↓H␈ε"random␈αsequence,␈α
using␈αour␈α
in␈α␈tuitiv␈α␈e␈αn␈α↓otion␈α
of␈α\truly␈α
random."␈α⊃One␈α
can␈αeasily
␈β¬T␈↓ ↓H␈ε"imagine␈α∂that␈α∂such␈α∂a␈α∂situation␈α∂will␈α∂ha␈α␈v␈α␈e␈α∂a␈α∂drastic␈α∂e{ect␈α∂if␈α∂this␈α∂set␈α∂of␈α∂a␈α∂million
␈β¬␈␈↓ ↓H␈ε"\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␈α
of␈αn␈α␈um␈α␈bers␈αfor␈αother␈αconceiv␈α}able␈αapplications␈αthat␈αuse␈α
extremely
␈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~W␈α⎇e␈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␈α
kn␈α↓o␈α␈ws␈α
the␈αmaterial␈α
ordinarily␈α
taugh␈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␈↓ λ1␈ε+n␈ε9!1
␈β∂B␈↓ εf␈∧∂Bεfα'
␈β∂H␈↓ ∧#␈ε+n␈ε9␈α␈!1
␈β∂t␈↓ αL␈ε↓␈␈↓ β!␈ε↓↓␈↓ λ→␈ε↓␈␈↓ λn␈ε↓↓␈↓ 	t␈ε↓␈␈↓ 
I␈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␈↓ πs␈ε"P␈↓ λ␈ε"r␈↓ λ'␈ε(S␈↓ λ@␈ε"(␈ε(n␈ε")␈↓ 	λ␈ε"and␈↓ 	N␈∧⊂∪	Nα'␈↓ 	N␈ε"P␈↓ 	f␈ε"r␈↓ 
α␈ε(S␈↓ 
≠␈ε"(␈ε(n␈ε")␈↓ 
W␈ε".
␈β⊂4␈↓ πs␈∧⊂4πsα'
␈β⊂B␈↓ α␈ε"W␈α⎇e␈α∂ha␈α␈v␈α␈e␈α∞seen␈α∂that␈α∂a␈↓ ∧d␈ε(k␈↓ ∧x␈ε"-distributed␈↓ ε>␈ε"[␈↓ εO␈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↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈βα"␈↓ ↓H␈ε2Theorem␈αA.␈ε/␈α→Let␈ε6␈αh␈↓ βk␈ε(U␈↓ ∧_␈ε6i␈ε"␈α
=␈↓ ∧\␈ε(U␈↓ ¬¬␈ε/,␈↓ ¬≠␈ε(U␈↓ ¬E␈ε/,␈↓ ¬[␈ε(U␈↓ ε∧␈ε/,␈↓ ε~␈ε".␈αε.␈αε.␈↓ εP␈ε/be␈αa␈↓ π≡␈ε"[␈↓ π/␈ε"0,␈αε1)␈ε/␈αsequence.␈α⊂If␈αthe␈αsequence
␈βα/␈↓ ∧∧␈ε+n␈↓ ∧u␈ε%0␈↓ ¬4␈ε%1␈↓ ¬t␈ε%2
␈βαr␈↓ ∧∂␈ε6hb␈↓ ∧)␈ε(b␈↓ ∧H␈ε(U␈↓ ∧u␈ε6ci␈ε"␈α
=␈ε6␈α
b␈↓ ¬U␈ε(b␈↓ ¬s␈ε(U␈↓ ε≥␈ε6c␈ε",␈ε6␈α∞b␈↓ εQ␈ε(b␈↓ εo␈ε(U␈↓ π→␈ε6c␈ε",␈ε6␈α∞b␈↓ πM␈ε(b␈↓ πk␈ε(U␈↓ λ∃␈ε6c␈ε",␈↓ λ;␈ε".␈αε.␈αε.
␈βα}␈↓ ∧9␈ε+j␈↓ ∧a␈ε+n␈↓ ¬d␈ε+j␈↓ ε␈ε%0␈↓ ε`␈ε+j␈↓ πλ␈ε%1␈↓ π\␈ε+j␈↓ λ∧␈ε%2
␈ββA␈↓ ↓H␈ε/is␈α∂a␈↓ α⊂␈ε(k␈↓ α%␈ε/-distributed␈↓ βl␈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␈↓ εp␈ε+j
␈ββl␈↓ ↓H␈ε"1␈α
<␈↓ α∩␈ε(b␈↓ α<␈ε"<␈↓ αj␈ε(b␈↓ β∀␈ε"<␈↓ βB␈ε(b␈↓ βl␈ε"<␈↓ ∧~␈ε6↓␈αε↓␈αε↓␈↓ ∧J␈ε/,␈αthen␈αthe␈αoriginal␈αsequence␈ε6␈αh␈↓ λ∩␈ε(U␈↓ λ?␈ε6i␈ε/␈αis␈↓ λ{␈ε(k␈↓ 	∂␈ε/-distributed.
␈ββx␈↓ α!␈ε%1␈↓ αy␈ε%2␈↓ βQ␈ε%3␈↓ λ+␈ε+n
␈β∧$␈↓ 	∃␈ε+j
␈β∧*␈↓ α␈ε"As␈α∪an␈α∪example␈α∪of␈α∪this␈α∀theorem,␈α∀suppose␈α∀that␈↓ λ∃␈ε(b␈↓ λI␈ε"=␈↓ 	β␈ε"2␈↓ 	$␈ε".␈α%The␈α∪sequence
␈β∧6␈↓ λ$␈ε+j
␈β∧O␈↓ ↓h␈ε+j␈↓ αi␈ε+j
␈β∧U␈↓ ↓H␈ε6b␈↓ ↓V␈ε"2␈↓ ↓w␈ε(U␈↓ α ␈ε6c␈ε",␈ε6␈α⊃b␈↓ αW␈ε"2␈↓ αx␈ε(U␈↓ β!␈ε6c␈ε",␈↓ βJ␈ε".␈αε.␈αε.␈↓ ∧β␈ε"is␈α∂essen␈α␈tially␈α⊂the␈α∂sequence␈α⊂of␈α∂the␈α∂|rst␈↓ λo␈ε(j␈↓ 	∂␈ε"bits␈α⊂of␈α∂the␈α∂binary
␈β∧a␈↓ α⊂␈ε%0␈↓ β⊃␈ε%1
␈β¬␈↓ ↓H␈ε"represen␈α␈tations␈α
of␈↓ βn␈ε(U␈↓ ∧↔␈ε",␈↓ ∧/␈ε(U␈↓ ∧Y␈ε",␈↓ ∧q␈ε".␈αε.␈αε.␈↓ ¬'␈ε".␈α∃If␈α
all␈α∞these␈α∞in␈α␈teger␈α
sequences␈α∞are␈↓ 	V␈ε(k␈↓ 	j␈ε"-distributed,
␈β¬␈↓ ∧π␈ε%0␈↓ ∧H␈ε%1
␈β¬+␈↓ ↓H␈ε"in␈αthe␈αsense␈αof␈αDe|nition␈αD␈↓ ∧t␈ε",␈αthen␈αthe␈αreal-v␈α}alued␈αsequence␈↓ λc␈ε(U␈↓ 	
␈ε",␈↓ 	#␈ε(U␈↓ 	L␈ε",␈↓ 	b␈ε".␈αε.␈αε.␈↓ 
_␈ε"m␈α␈ust␈αalso
␈β¬8␈↓ λ|␈ε%0␈↓ 	<␈ε%1
␈β¬V␈↓ ↓H␈ε"be␈↓ ↓x␈ε(k␈↓ α␈ε"-distributed␈αin␈αthe␈αsense␈αof␈αDe|nition␈αB.
␈βε∀␈↓ ↓H␈ε/Pro␈α↓of␈α
of␈αTheorem␈αA.␈ε"␈α∀If␈αthe␈α
sequence␈ε6␈αb␈ε(b␈↓ ε0␈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!␈↓ εI␈ε%0␈↓ π3␈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∪No␈α␈w␈α∞let␈↓ π0␈ε(u␈↓ πT␈ε",␈↓ πk␈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␈↓ βI␈ε(u␈↓ βw␈ε6∀␈↓ ∧%␈ε(u␈↓ ∧S␈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␈↓ ∧:␈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␈↓ ¬G␈ε(u␈↓ ¬x␈ε6∀␈↓ ε)␈ε(U␈↓ εb␈ε"<␈↓ π∩␈ε(v␈↓ π4␈ε",␈↓ πK␈ε".␈αε.␈αε.␈↓ π{␈ε",␈↓ λ∪␈ε(u␈↓ λF␈ε6∀␈↓ λv␈ε(U␈↓ 

␈ε"<␈↓ 
:␈ε(v␈↓ 
↑␈ε".␈α∪W␈α⎇e
␈βλE␈↓ ¬\␈ε%1␈↓ εA␈ε+n␈↓ π#␈ε%1␈↓ λ(␈ε+k␈↓ 	∂␈ε+n␈ε%␈α␈+␈↓ 	?␈ε+k␈↓ 	P␈ε9␈␈ε%␈α␈1␈↓ 
L␈ε+k
␈βλc␈↓ ↓H␈ε"ha␈α␈v␈α␈e
␈β	!␈↓ α∩␈ε↓␈␈↓ αg␈ε↓↓
␈β	&␈↓ βY␈ε↓∩␈↓ 
→␈ε↓∪
␈β	'␈↓ ε
␈ε"1␈↓ 
β␈ε"1
␈β	7␈↓ ∧∧␈ε90␈↓ ¬D␈ε90␈↓ π≤␈ε90␈↓ 	9␈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∀␈↓ ∧M␈ε(U␈↓ ¬∧␈ε"<␈↓ ¬2␈ε(v␈↓ ¬]␈ε"+␈↓ ε#␈ε",␈↓ ε=␈ε".␈αε.␈αε.␈↓ εm␈ε",␈↓ ππ␈ε(u␈↓ π8␈ε6∀␈↓ πf␈ε(U␈↓ λx␈ε"<␈↓ 	&␈ε(v␈↓ 	S␈ε"+
␈β	@␈↓ ↓l␈∧	@↓lα'
␈β	L␈↓ ∧e␈ε+n␈↓ π␈␈ε+n␈ε%+␈↓ λ/␈ε+k␈↓ λ@␈ε9␈␈ε%␈α␈1
␈β	P␈↓ ∧∧␈ε%1␈↓ ¬D␈ε%1␈↓ π≤␈ε+k␈↓ 	9␈ε+k
␈β	S␈↓ ε
␈∧	Sε
α∩␈↓ 
β␈∧	S
βα∩
␈β	Y␈↓ ε∂␈ε(b␈↓ 
¬␈ε(b
␈β
∧␈↓ εW␈ε↓∩␈↓ λ8␈ε↓∪␈↓ 	∧␈ε↓∩␈↓ 
h␈ε↓∪
␈β
¬␈↓ λ"␈ε"1␈↓ 
R␈ε"1
␈β
∃␈↓ π␈ε90␈↓ πZ␈ε90␈↓ 	-␈ε90␈↓ 
λ␈ε90
␈β
≥␈↓ ε)␈ε"=␈↓ εm␈ε(v␈↓ π→␈ε6␈␈↓ πE␈ε(u␈↓ πr␈ε"+␈↓ λT␈ε".␈αε.␈αε.␈↓ 	~␈ε(v␈↓ 	H␈ε6␈␈↓ 	t␈ε(u␈↓ 
"␈ε"+␈↓ 
}␈ε";
␈β
.␈↓ π␈ε%1␈↓ πZ␈ε%1␈↓ 	-␈ε+k␈↓ 
λ␈ε+k
␈β
1␈↓ λ"␈∧
1λ"α∩␈↓ 
R␈∧
1
Rα∩
␈β
7␈↓ λ$␈ε(b␈↓ 
T␈ε(b
␈β
i␈↓ α∩␈ε↓␈␈↓ αg␈ε↓↓
␈β
n␈↓ βY␈ε↓∩␈↓ 
→␈ε↓∪
␈β
o␈↓ ∧M␈ε"1␈↓ πf␈ε"1
␈β
␈␈↓ ∧∧␈ε90␈↓ ε∩␈ε90␈↓ π≤␈ε90␈↓ 
π␈ε90
␈βπ␈↓ ↓l␈ε"P␈↓ α∧␈ε"r␈↓ α ␈ε(S␈↓ α9␈ε"(␈ε(n␈ε")␈↓ α␈␈ε6∃␈↓ β-␈ε"P␈↓ βE␈ε"r␈↓ βo␈ε(u␈↓ ∧≥␈ε"+␈↓ ∧m␈ε6∀␈↓ ¬≠␈ε(U␈↓ ¬R␈ε"<␈↓ ε␈ε(v␈↓ ε#␈ε",␈↓ ε=␈ε".␈αε.␈αε.␈↓ εm␈ε",␈↓ ππ␈ε(u␈↓ π6␈ε"+␈↓ λε␈ε6∀␈↓ λ4␈ε(U␈↓ 	F␈ε"<␈↓ 	t␈ε(v
␈β∀␈↓ ¬3␈ε+n␈↓ λM␈ε+n␈ε%+␈↓ λ⎇␈ε+k␈↓ 	∞␈ε9␈␈ε%␈α␈1
␈β_␈↓ ∧∧␈ε%1␈↓ ε∩␈ε%1␈↓ π≤␈ε+k␈↓ 
π␈ε+k
␈β≠␈↓ ∧M␈∧≠∧Mα∩␈↓ πf␈∧≠πfα∩
␈β!␈↓ ∧N␈ε(b␈↓ πh␈ε(b
␈β)␈↓ ↓l␈∧)↓lα'
␈βL␈↓ εW␈ε↓∩␈↓ λ8␈ε↓∪␈↓ 	∧␈ε↓∩␈↓ 
h␈ε↓∪
␈βM␈↓ λ"␈ε"1␈↓ 
R␈ε"1
␈β]␈↓ π␈ε90␈↓ πZ␈ε90␈↓ 	-␈ε90␈↓ 
λ␈ε90
␈βe␈↓ ε)␈ε"=␈↓ εm␈ε(v␈↓ π→␈ε6␈␈↓ πE␈ε(u␈↓ πr␈ε6␈␈↓ λT␈ε".␈αε.␈αε.␈↓ 	~␈ε(v␈↓ 	H␈ε6␈␈↓ 	t␈ε(u␈↓ 
"␈ε6␈␈↓ 
}␈ε".
␈βv␈↓ π␈ε%1␈↓ πZ␈ε%1␈↓ 	-␈ε+k␈↓ 
λ␈ε+k
␈βy␈↓ λ"␈∧yλ"α∩␈↓ 
R␈∧y
Rα∩
␈β␈␈↓ λ$␈ε(b␈↓ 
T␈ε(b
␈βD␈↓ αG␈ε90␈↓ β$␈ε90
␈βJ␈↓ ↓H␈ε"No␈α␈w␈ε6␈α∂j␈ε"(␈↓ α4␈ε(v␈↓ αa␈ε6␈␈↓ β∂␈ε(u␈↓ β>␈ε6ε␈ε"␈α
1/␈ε(b␈ε")␈ε6␈α␈␈ε"␈α
(␈↓ ∧p␈ε(v␈↓ ¬≤␈ε6␈␈↓ ¬J␈ε(u␈↓ ¬n␈ε")␈ε6j␈α⊂∀␈ε"␈α⊂2/␈ε(b␈ε";␈α∩since␈α⊂our␈α∂inequalities␈α⊂h␈α↓old␈α⊂for␈α∂all
␈βV␈↓ ¬α␈ε+j␈↓ ¬←␈ε+j
␈β[␈↓ αG␈ε+j␈↓ β$␈ε+j
␈βu␈↓ ↓H␈ε(b␈ε"␈α
=␈↓ α∂␈ε(b␈↓ α-␈ε",␈αand␈αsince␈↓ βa␈ε(b␈↓ ∧
␈ε6!␈α
1␈ε"␈αas␈↓ ¬∀␈ε(j␈↓ ¬/␈ε6!␈α
1␈ε",␈αw␈α␈e␈αha␈α␈v␈α␈e
␈β
↓␈↓ α≡␈ε+j␈↓ βq␈ε+j
␈β
*␈↓ ¬→␈ε↓␈␈↓ ¬n␈ε↓↓␈↓ ε[␈ε↓␈␈↓ π/␈ε↓↓
␈β
H␈↓ ↓[␈ε"(␈↓ ↓g␈ε(v␈↓ α⊃␈ε6␈␈↓ α=␈ε(u␈↓ αc␈ε")␈↓ αu␈ε".␈αε.␈αε.␈↓ β%␈ε"(␈↓ β1␈ε(v␈↓ β\␈ε6␈␈↓ ∧λ␈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␈␈↓ λX␈ε(u␈↓ λ⎇␈ε")␈↓ 	∂␈ε".␈αε.␈αε.␈↓ 	?␈ε"(␈↓ 	K␈ε(v␈↓ 	w␈ε6␈␈↓ 
#␈ε(u␈↓ 
J␈ε").
␈β
Q␈↓ π␈∧
Qπ≠∂
␈β
U␈↓ ↓x␈ε%1␈↓ αR␈ε%1␈↓ βB␈ε+k␈↓ ∧≥␈ε+k␈↓ λ∪␈ε%1␈↓ λl␈ε%1␈↓ 	]␈ε+k␈↓ 
8␈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␈α
h␈↓ β↑␈ε(U␈↓ ∧␈ε6i␈ε/␈α
be␈αa␈↓ ∧l␈ε(k␈↓ ¬␈ε/-distributed␈↓ εB␈ε"[␈↓ εR␈ε"0,␈αε1)␈ε/␈αsequence,␈α
and␈α
let␈↓ 	/␈ε(f␈↓ 	E␈ε"(␈↓ 	Q␈ε(x␈↓ 	u␈ε",␈↓ 
¬␈ε(x␈↓ 
*␈ε",␈↓ 
:␈ε".␈αε.␈αε.␈↓ 
j␈ε",␈↓ 
z␈ε(x␈↓  ␈ε")
␈β∂
␈↓ βw␈ε+n␈↓ 	e␈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;␈↓ πN␈ε/then
␈β∂g␈↓ ∧f␈ε↓X
␈β∂q␈↓ ∧,␈ε"1
␈β⊂	␈↓ β`␈ε"lim␈↓ ¬<␈ε(f␈↓ ¬R␈ε"(␈↓ ¬↑␈ε(U␈↓ εε␈ε",␈↓ ε⊗␈ε(U␈↓ εj␈ε",␈↓ εz␈ε".␈αε.␈αε.␈↓ π*␈ε",␈↓ π:␈ε(U␈↓ λ=␈ε")
␈β⊂⊗␈↓ ¬v␈ε+j␈↓ ε.␈ε+j␈↓ ε<␈ε%+1␈↓ πS␈ε+j␈↓ πa␈ε%+␈↓ π}␈ε+k␈↓ λ∂␈ε9␈␈ε%␈α␈1
␈β⊂≡␈↓ ∧*␈∧⊂≡∧*α⊗
␈β⊂#␈↓ ∧*␈ε(n
␈β⊂*␈↓ βQ␈ε+n␈ε9!␈α␈1
␈β⊂:␈↓ ∧J␈ε%0␈ε9∀␈↓ ∧w␈ε+j␈↓ ¬∧␈ε%<␈ε+n
␈β⊂F␈↓ α␈ε"(8)
␈β⊂W␈↓ ∧K␈ε↓Z␈↓ ¬<␈ε↓Z
␈β⊂]␈↓ ∧o␈ε%1␈↓ ¬`␈ε%1
␈β⊃β␈↓ ∧≥␈ε"=␈↓ ¬ε␈ε6↓␈αε↓␈αε↓␈↓ ¬w␈ε(f␈↓ ε␈ε"(␈↓ ε_␈ε(x␈↓ ε=␈ε",␈↓ εM␈ε(x␈↓ εr␈ε",␈↓ πα␈ε".␈αε.␈αε.␈↓ π2␈ε",␈↓ πB␈ε(x␈↓ πh␈ε")␈ε(␈αεd␈↓ λ␈ε(x␈↓ λ7␈ε".␈αε.␈αε.␈↓ λg␈ε(d␈↓ λy␈ε(x␈↓ 	∨␈ε".
␈β⊃⊂␈↓ ε,␈ε%1␈↓ εa␈ε%2␈↓ πV␈ε+k␈↓ λ ␈ε%1␈↓ 	
␈ε+k
␈β⊃&␈↓ ∧←␈ε%0␈↓ ¬P␈ε%0
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"147
␈β↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈βα$␈↓ ↓H␈ε/Pro␈α↓of.␈ε"␈α~The␈α
de|nition␈α
of␈α
a␈↓ ∧y␈ε(k␈↓ ¬
␈ε"-distributed␈α
sequence␈αstates␈α
that␈α
this␈α
result␈α
is␈αtrue
␈βαO␈↓ ↓H␈ε"in␈αthe␈αspecial␈αcase␈αthat
␈ββ~␈↓ ∧u␈ε"1,␈↓ ¬Y␈ε"if␈↓ ¬{␈ε(u␈↓ ε*␈ε6∀␈↓ εX␈ε(x␈↓ ππ␈ε"<␈↓ π5␈ε(v␈↓ πW␈ε",␈↓ πg␈ε".␈αε.␈αε.␈↓ λ↔␈ε",␈↓ λ'␈ε(u␈↓ λX␈ε6∀␈↓ 	ε␈ε(x␈↓ 	6␈ε"<␈↓ 	d␈ε(v␈↓ 
λ␈ε";
␈ββ≤␈↓ ∧←␈ε↓~
␈ββ'␈↓ ε∂␈ε%1␈↓ εl␈ε%1␈↓ πF␈ε%1␈↓ λ<␈ε+k␈↓ 	~␈ε+k␈↓ 	v␈ε+k
␈ββ5␈↓ α↑␈ε(f␈↓ αt␈ε"(␈↓ β␈ε(x␈↓ β$␈ε",␈↓ β4␈ε".␈αε.␈αε.␈↓ βd␈ε",␈↓ βt␈ε(x␈↓ ∧≠␈ε")␈α
=␈↓ α␈ε"(9)
␈ββB␈↓ β∀␈ε%1␈↓ ∧λ␈ε+k
␈ββP␈↓ ∧u␈ε"0,␈↓ ¬Y␈ε"otherwise.
␈β∧≡␈↓ ↓H␈ε"Therefore␈α	Eq.␈α	(8)␈α
is␈α	true␈α
whenev␈α␈er␈↓ ¬`␈ε(f␈↓ ¬␈␈ε"=␈↓ ε-␈ε(a␈↓ εQ␈ε(f␈↓ εw␈ε"+␈↓ π≥␈ε(a␈↓ πA␈ε(f␈↓ πg␈ε"+␈↓ λ
␈ε6↓␈αε↓␈αε↓␈↓ λ:␈ε"+␈↓ λa␈ε(a␈↓ 	⊂␈ε(f␈↓ 	H␈ε"and␈α	when␈α	each
␈β∧*␈↓ ε@␈ε%1␈↓ εc␈ε%1␈↓ π0␈ε%2␈↓ πS␈ε%2␈↓ λt␈ε+m␈↓ 	"␈ε+m
␈β∧I␈↓ ↓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∧U␈↓ ↓Y␈ε+j
␈β∧t␈↓ ↓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¬K␈↓ ↓H␈ε"v␈α}alue␈αto␈↓ αQ␈ε(f␈↓ αr␈ε"on␈αeach␈αsubcell.
␈β¬v␈↓ 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-
␈βεL␈↓ ↓H␈ε"tions␈↓ α∨␈ε(f␈↓ α?␈ε"and␈↓ β∧␈∧εLβ∧α⊗␈↓ β∧␈ε(f␈↓ β%␈ε"such␈αthat␈↓ ∧C␈ε(f␈↓ ∧X␈ε"(␈↓ ∧d␈ε(x␈↓ ¬	␈ε",␈↓ ¬→␈ε".␈αε.␈αε.␈↓ ¬I␈ε",␈↓ ¬Y␈ε(x␈↓ ¬␈␈ε")␈ε6␈α
∀␈↓ εC␈ε(f␈↓ εY␈ε"(␈↓ εe␈ε(x␈↓ π
␈ε",␈↓ π~␈ε".␈αε.␈αε.␈↓ πJ␈ε",␈↓ πZ␈ε(x␈↓ λ␈ε")␈ε6␈α
∀␈↓ λD␈∧εLλDα⊗␈↓ λD␈ε(f␈↓ λY␈ε"(␈↓ λe␈ε(x␈↓ 	
␈ε",␈↓ 	~␈ε".␈αε.␈αε.␈↓ 	J␈ε",␈↓ 	Z␈ε(x␈↓ 
␈ε"),␈αand␈αsuch
␈βεY␈↓ ∧x␈ε%1␈↓ ¬m␈ε+k␈↓ εy␈ε%1␈↓ πn␈ε+k␈↓ λy␈ε%1␈↓ 	n␈ε+k
␈βεu␈↓ α∨␈∧εuα∨α⊗␈↓ ∧C␈∧εu∧Cα⊗
␈βε⎇␈↓ ↓H␈ε"that␈αthe␈αdi{erence␈αof␈αthe␈α
in␈α␈tegrals␈αof␈↓ ε_␈ε(f␈↓ ε9␈ε"and␈↓ π␈∧ε⎇πα⊗␈↓ π␈ε(f␈↓ π!␈ε"is␈αless␈αthan␈ε(␈α
∂␈ε"␈α␈.␈α⊃Since␈αEq.␈α(8)␈αh␈α↓olds
␈βπ&␈↓ ε_␈∧π&ε_α⊗
␈βπ.␈↓ αg␈∧π.αgα⊗
␈βπ/␈↓ ↓H␈ε"for␈↓ α␈ε(f␈↓ α!␈ε"and␈↓ αg␈ε(f␈↓ α|␈ε",␈αand␈αsince
␈βπW␈↓ α␈∧πWαα⊗
␈βπt␈↓ β≠␈ε↓X␈↓ π⊃␈ε↓X
␈βπ␈␈↓ 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")
␈βλ#␈↓ ∧+␈ε+j␈↓ ¬#␈ε+j␈↓ ¬0␈ε%+␈↓ ¬N␈ε+k␈↓ ¬←␈ε9␈␈ε%␈α␈1␈↓ λ!␈ε+j␈↓ 	→␈ε+j␈↓ 	'␈ε%+␈↓ 	D␈ε+k␈↓ 	U␈ε9␈␈ε%␈α␈1
␈βλ+␈↓ α←␈∧λ+α←α⊗␈↓ εU␈∧λ+εUα⊗
␈βλ0␈↓ α←␈ε(n␈↓ εU␈ε(n
␈βλ?␈↓ βq␈∧λ?βqα⊗
␈βλG␈↓ α␈␈ε%0␈ε9∀␈↓ β+␈ε+j␈↓ β9␈ε%<␈ε+n␈↓ εu␈ε%0␈ε9∀␈↓ π"␈ε+j␈↓ π/␈ε%<␈ε+n
␈βλ`␈↓ π⊃␈ε↓X
␈βλj␈↓ εW␈ε"1
␈β	α␈↓ ε#␈ε6∀␈↓ πg␈∧	απgα⊗␈↓ 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%+␈↓ 	D␈ε+k␈↓ 	U␈ε9␈␈ε%␈α␈1
␈β	↔␈↓ εU␈∧	↔εUα⊗
␈β	≤␈↓ εU␈ε(n
␈β	3␈↓ εu␈ε%0␈ε9∀␈↓ π"␈ε+j␈↓ π/␈ε%<␈ε+n
␈β	z␈↓ ↓H␈ε"w␈α␈e␈αconclude␈αthat␈αEq.␈α(8)␈αis␈αtrue␈αalso␈αfor␈↓ εE␈ε(f␈↓ εZ␈ε".
␈β
β␈↓ π␈∧
βπ≠∂
␈β
9␈↓ α␈ε"Theorem␈αB␈αcan␈αbe␈αapplied,␈α
for␈αexample,␈αto␈αthe␈↓ π⎇␈ε/perm␈α␈utation␈αtest␈ε"␈αof␈αSection
␈β
e␈↓ ↓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␈↓ ∧M␈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
q␈↓ β∃␈ε%1␈↓ βH␈ε%2␈↓ ∧:␈ε+k
␈β⊂␈↓ ↓H␈ε"w␈α␈an␈α␈t␈αto␈αsh␈α↓o␈α␈w␈αthat
␈βc␈↓ β⊂␈ε"P␈↓ β(␈ε"r␈↓ β6␈ε"(␈↓ βB␈ε(U␈↓ ∧a␈ε"<␈↓ ¬∂␈ε(U␈↓ ε.␈ε"<␈↓ ε\␈ε6↓␈αε↓␈αε↓␈↓ π⊂␈ε"<␈↓ π>␈ε(U␈↓ λT␈ε")␈α
=␈α
1/␈↓ 	<␈ε(k␈↓ 	P␈ε"!.␈↓ 
p␈ε"(10)
␈βo␈↓ β[␈ε+n␈ε%+␈↓ ∧␈ε+p␈↓ ∧)␈ε9␈␈ε%␈α␈1␈↓ ¬(␈ε+n␈ε%+␈↓ ¬X␈ε+p␈↓ ¬v␈ε9␈␈ε%␈α␈1␈↓ πW␈ε+n␈ε%+␈↓ λπ␈ε+p␈↓ λ&␈ε9␈␈ε%␈α␈1
␈βy␈↓ ∧~␈ε'1␈↓ ¬g␈ε'2␈↓ λ⊗␈ε-k
␈β5␈↓ ↓H␈ε"T␈α⎇o␈αpro␈α␈v␈α␈e␈αthis,␈αassume␈αthat␈αthe␈αsequence␈ε6␈αh␈↓ εV␈ε(U␈↓ πβ␈ε6i␈ε"␈αis␈↓ π?␈ε(k␈↓ πS␈ε"-distributed,␈αand␈αlet
␈βB␈↓ εn␈ε+n
␈β
α␈↓ ¬E␈ε"1,␈↓ ε)␈ε"if␈↓ εK␈ε(x␈↓ πλ␈ε"<␈↓ π6␈ε(x␈↓ πs␈ε"<␈↓ λ!␈ε6↓␈αε↓␈αε↓␈↓ λU␈ε"<␈↓ 	β␈ε(x␈↓ 	8␈ε";
␈β
∧␈↓ ¬/␈ε↓~
␈β
∂␈↓ ε←␈ε+p␈↓ πJ␈ε+p␈↓ 	↔␈ε+p
␈β
_␈↓ εn␈ε'1␈↓ πY␈ε'2␈↓ 	'␈ε-k
␈β
≥␈↓ β/␈ε(f␈↓ βD␈ε"(␈↓ βP␈ε(x␈↓ βu␈ε",␈↓ ∧¬␈ε".␈αε.␈αε.␈↓ ∧5␈ε",␈↓ ∧E␈ε(x␈↓ ∧k␈ε")␈α
=
␈β
*␈↓ βd␈ε%1␈↓ ∧Y␈ε+k
␈β
8␈↓ ¬E␈ε"0,␈↓ ε)␈ε"otherwise.
␈β∞∧␈↓ ↓H␈ε"W␈α⎇e␈αha␈α␈v␈α␈e
␈β∞J␈↓ β⊗␈ε"P␈↓ β.␈ε"r␈↓ β<␈ε"(␈↓ βH␈ε(U␈↓ ∧g␈ε"<␈↓ ¬∃␈ε(U␈↓ ε4␈ε"<␈↓ εb␈ε6↓␈αε↓␈αε↓␈↓ π⊗␈ε"<␈↓ πD␈ε(U␈↓ λZ␈ε")
␈β∞V␈↓ βa␈ε+n␈ε%␈α␈+␈↓ ∧⊃␈ε+p␈↓ ∧/␈ε9␈␈ε%␈α␈1␈↓ ¬.␈ε+n␈ε%␈α␈+␈↓ ¬↑␈ε+p␈↓ ¬|␈ε9␈␈ε%␈α␈1␈↓ π]␈ε+n␈ε%␈α␈+␈↓ λ
␈ε+p␈↓ λ,␈ε9␈␈ε%␈α␈1
␈β∞`␈↓ ∧ ␈ε'1␈↓ ¬m␈ε'2␈↓ λ≤␈ε-k
␈β∞{␈↓ ∧⊂␈ε↓Z␈↓ ¬↓␈ε↓Z
␈β∂↓␈↓ ∧4␈ε%1␈↓ ¬%␈ε%1
␈β∂'␈↓ βb␈ε"=␈↓ ∧K␈ε6↓␈αε↓␈αε↓␈↓ ¬;␈ε(f␈↓ ¬Q␈ε"(␈↓ ¬]␈ε(x␈↓ ε↓␈ε",␈↓ ε⊃␈ε".␈αε.␈αε.␈↓ εA␈ε",␈↓ εQ␈ε(x␈↓ εx␈ε")␈ε(␈αεd␈↓ π≤␈ε(x␈↓ πG␈ε".␈αε.␈αε.␈↓ πw␈ε(d␈↓ λ	␈ε(x
␈β∂3␈↓ ¬q␈ε%1␈↓ εe␈ε+k␈↓ π0␈ε%1␈↓ λ≥␈ε+k
␈β∂J␈↓ ∧$␈ε%0␈↓ ¬∃␈ε%0
␈β∂←␈↓ ∧⊂␈ε↓Z␈↓ ¬≥␈ε↓Z␈↓ ε-␈ε↓Z␈↓ πW␈ε↓Z
␈β∂e␈↓ ∧4␈ε%1␈↓ ¬A␈ε+x␈↓ εQ␈ε+x␈↓ π{␈ε+x
␈β∂n␈↓ ¬Q␈ε-p␈↓ εa␈ε-p␈↓ λ␈ε-p
␈β∂s␈↓ εo␈ε'3␈↓ λ→␈ε'2␈↓ 	>␈ε"1
␈β∂t␈↓ ¬←␈ε-k
␈β⊂␈↓ βb␈ε"=␈↓ ∧Q␈ε(d␈↓ ∧c␈ε(x␈↓ ¬w␈ε6↓␈αε↓␈αε↓␈↓ π␈ε(d␈↓ π≡␈ε(x␈↓ λ7␈ε(d␈↓ λI␈ε(x␈↓ 	ε␈ε"=␈↓ 	Z␈ε".
␈β⊂↔␈↓ ∧w␈ε+p␈↓ π2␈ε+p␈↓ λ]␈ε+p
␈β⊂∨␈↓ 	8␈∧⊂∨	8α∨
␈β⊂!␈↓ ¬ε␈ε-k␈↓ πA␈ε'2␈↓ λl␈ε'1
␈β⊂$␈↓ 	8␈ε(k␈↓ 	L␈ε"!
␈β⊂.␈↓ ∧$␈ε%0␈↓ ¬1␈ε%0␈↓ εA␈ε%0␈↓ πk␈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↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈βα$␈↓ α␈ε"W␈α⎇e␈αcan␈αalso␈αsh␈α↓o␈α␈w␈αthat␈αthe␈↓ ¬9␈ε/serial␈αcorrelation␈αtest␈ε"␈αis␈αsatis|ed:
␈βαg␈↓ ↓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␈↓ ∧2␈ε/and␈↓ ∧x␈ε(U␈↓ ¬←␈ε/tends␈αto␈αzero:
␈ββ≡␈↓ ∧⊃␈ε+n␈↓ ¬⊂␈ε+n␈ε%+␈↓ ¬A␈ε+k
␈ββM␈↓ ∧8␈ε↓P␈↓ εC␈ε↓P␈↓ πP␈ε↓P
␈ββc␈↓ ∧≥␈ε%1␈↓ ε(␈ε%1␈↓ π5␈ε%1
␈ββh␈↓ ∧d␈ε(U␈↓ ¬␈ε(U␈↓ ¬j␈ε6␈␈↓ ε⊗␈ε"(␈↓ εo␈ε(U␈↓ π↔␈ε")␈↓ π#␈ε"(␈↓ π|␈ε(U␈↓ λR␈ε")
␈ββt␈↓ ∧⎇␈ε+j␈↓ ¬%␈ε+j␈↓ ¬3␈ε%+␈↓ ¬P␈ε+k␈↓ πλ␈ε+j␈↓ λ∀␈ε+j␈↓ λ"␈ε%+␈↓ λ@␈ε+k
␈ββx␈↓ ∧≤␈ε+n␈↓ ε&␈ε+n␈↓ π3␈ε+n
␈ββ|␈↓ ∧≤␈∧β|∧≤α∪␈↓ ε&␈∧β|ε&α∪␈↓ π3␈∧β|π3α∪
␈β∧↓␈↓ αG␈ε"lim␈↓ β∩␈ε↓q␈↓ 	r␈ε"=␈α
0.
␈β∧⊂␈↓ β6␈ε↓␈␈↓ ∧r␈ε↓␈␈↓ ¬u␈ε↓↓␈↓ ε∀␈ε↓↓␈↓ ε"␈ε↓␈␈↓ λε␈ε↓␈␈↓ 	7␈ε↓↓␈↓ 	V␈ε↓↓
␈β∧∀␈↓ βd␈ε↓P␈↓ ¬!␈ε↓P␈↓ εP␈ε↓P␈↓ λ5␈ε↓P
␈β∧∃␈↓ β∩␈∧∧∃β∩αεR
␈β∧∨␈↓ β6␈∧∧∨β6αε.
␈β∧ ␈↓ εβ␈ε%2␈↓ 	E␈ε%2
␈β∧!␈↓ α8␈ε+n␈ε9!1
␈β∧*␈↓ βI␈ε%1␈↓ ∧.␈ε%2␈↓ ¬ε␈ε%1␈↓ ε5␈ε%1␈↓ π~␈ε%2␈↓ λ~␈ε%1
␈β∧/␈↓ ∧⊂␈ε(U␈↓ ∧F␈ε6␈␈↓ ¬M␈ε(U␈↓ ε|␈ε(U␈↓ πZ␈ε6␈␈↓ λa␈ε(U
␈β∧;␈↓ ¬f␈ε+j␈↓ λz␈ε+j␈↓ 	π␈ε%+␈↓ 	%␈ε+k
␈β∧?␈↓ βH␈ε+n␈↓ ¬∧␈ε+n␈↓ ε4␈ε+n␈↓ λ_␈ε+n
␈β∧A␈↓ ∧)␈ε+j␈↓ π∃␈ε+j␈↓ π"␈ε%+␈↓ π@␈ε+k
␈β∧C␈↓ βH␈∧∧CβHα∪␈↓ ¬∧␈∧∧C¬∧α∪␈↓ ε4␈∧∧Cε4α∪␈↓ λ_␈∧∧Cλ_α∪
␈β¬
␈↓ ↓H␈ε"(All␈αsummations␈αhere␈αare␈αfor␈α0␈ε6␈α
∀␈↓ ¬b␈ε(j␈↓ ¬⎇␈ε"<␈ε(␈α
n␈ε".)
␈β¬Q␈↓ ↓H␈ε/Pro␈α↓of.␈ε"␈α→By␈αTheorem␈αB␈↓ ∧)␈ε",␈αthe␈αquan␈α␈tities
␈βε
␈↓ α+␈ε↓P␈↓ ∧N␈ε↓P␈↓ ε ␈ε↓P␈↓ λ≠␈ε↓P␈↓ 	g␈ε↓P
␈βε"␈↓ ¬↔␈ε%2␈↓ εj␈ε%2
␈βε#␈↓ α⊂␈ε%1␈↓ ∧3␈ε%1␈↓ εε␈ε%1␈↓ λ␈ε%1␈↓ 	M␈ε%1
␈βε(␈↓ αW␈ε(U␈↓ α␈␈ε(U␈↓ βU␈ε",␈↓ ∧z␈ε(U␈↓ ¬(␈ε",␈↓ εL␈ε(U␈↓ π"␈ε",␈↓ λG␈ε(U␈↓ λo␈ε",␈↓ 
∪␈ε(U
␈βε4␈↓ αp␈ε+j␈↓ β_␈ε+j␈↓ β%␈ε%+␈↓ βC␈ε+k␈↓ λ`␈ε+j␈↓ 
,␈ε+j␈↓ 
:␈ε%+␈↓ 
W␈ε+k
␈βε9␈↓ α∂␈ε+n␈↓ ∧1␈ε+n␈↓ ¬∩␈ε+j␈↓ ε∧␈ε+n␈↓ εe␈ε+j␈↓ εr␈ε%+␈↓ π⊂␈ε+k␈↓ π}␈ε+n␈↓ 	K␈ε+n
␈βε<␈↓ α∂␈∧ε<α∂α∪␈↓ ∧1␈∧ε<∧1α∪␈↓ ε∧␈∧ε<ε∧α∪␈↓ π}␈∧ε<π}α∪␈↓ 	K␈∧ε<	Kα∪
␈βε{␈↓ ¬∩␈ε%1␈↓ ¬?␈ε%1␈↓ ¬l␈ε%1␈↓ ε→␈ε%1␈↓ εG␈ε%1
␈βε␈␈↓ ↓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".
␈βπλ␈↓ λ5␈∧πλλ5≠∂
␈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⊂
␈βπB␈↓ 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
␈βπm␈↓ ↓H␈ε"sequences.␈α~W␈α⎇e␈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
␈βλD␈↓ ↓H␈ε"poin␈α␈ts
␈βλp␈↓ β5␈ε"(␈↓ βA␈ε(U␈↓ βk␈ε",␈↓ β{␈ε(U␈↓ ∧$␈ε"),␈α∞(␈↓ ∧T␈ε(U␈↓ ∧}␈ε",␈↓ ¬∞␈ε(U␈↓ ¬7␈ε"),␈α∞(␈↓ ¬g␈ε(U␈↓ ε⊃␈ε",␈↓ ε!␈ε(U␈↓ εK␈ε"),␈α∞(␈↓ ε{␈ε(U␈↓ π$␈ε",␈↓ π4␈ε(U␈↓ π↑␈ε"),␈α∞(␈↓ λ∞␈ε(U␈↓ λ7␈ε",␈↓ λG␈ε(U␈↓ λq␈ε"),␈↓ 	∃␈ε".␈αε.␈αε.
␈βλ|␈↓ βZ␈ε%0␈↓ ∧∀␈ε%1␈↓ ∧m␈ε%1␈↓ ¬'␈ε%2␈↓ ε␈ε%2␈↓ ε:␈ε%3␈↓ π∪␈ε%3␈↓ πM␈ε%4␈↓ λ&␈ε%4␈↓ λ`␈ε%5
␈β	5␈↓ ↓H␈ε"are␈α
equidistributed␈α
in␈αthe␈α
unit␈α
square.␈α⊂It␈α
is␈α
quite␈αpossible,␈α
h␈α↓o␈α␈w␈α␈ev␈α␈er,␈αthat␈α
this␈α
can
␈β	`␈↓ ↓H␈ε"happen␈α∂while␈α⊂alternate␈α∂pairs␈α⊂of␈α∂poin␈α␈ts␈α⊂(␈↓ εJ␈ε(U␈↓ εs␈ε",␈↓ πβ␈ε(U␈↓ π-␈ε"),␈α⊂(␈↓ π←␈ε(U␈↓ λ	␈ε",␈↓ λ→␈ε(U␈↓ λB␈ε"),␈α⊃(␈↓ λu␈ε(U␈↓ 	≡␈ε",␈↓ 	.␈ε(U␈↓ 	X␈ε"),␈↓ 	}␈ε".␈αε.␈αε.␈↓ 
8␈ε"are␈ε/␈α∂n␈α↓ot
␈β	l␈↓ εc␈ε%1␈↓ π≤␈ε%2␈↓ πx␈ε%3␈↓ λ2␈ε%4␈↓ 	∞␈ε%5␈↓ 	G␈ε%6
␈β
␈↓ ↓H␈ε"equidistributed;␈α∞if␈α∞the␈α∞densit␈α␈y␈α∞of␈α∞poin␈α␈ts␈α∞(␈↓ εS␈ε(U␈↓ π<␈ε",␈↓ πL␈ε(U␈↓ λλ␈ε")␈α∞is␈α∞de|cien␈α␈t␈α∞in␈α∞some␈α
area,
␈β
_␈↓ εl␈ε%2␈ε+n␈ε9␈α␈␈␈ε%1␈↓ πe␈ε%2␈ε+n
␈β
6␈↓ ↓H␈ε"the␈α⊃other␈α⊃poin␈α␈ts␈α⊃(␈↓ βj␈ε(U␈↓ ∧&␈ε",␈↓ ∧6␈ε(U␈↓ ¬∨␈ε")␈α∩migh␈α␈t␈α⊃compensate.␈α∨F␈α⎇or␈α∩example,␈α∩the␈α⊃periodic
␈β
C␈↓ ∧β␈ε%2␈ε+n␈↓ ∧O␈ε%2␈ε+n␈ε%+1
␈β
b␈↓ ↓H␈ε"binary␈αsequence
␈β9␈↓ α[␈ε6h␈↓ αg␈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)
␈βE␈↓ β¬␈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␈↓ ∧R␈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
␈βH␈↓ ∧.␈ε%2␈ε+␈α↓n
␈βf␈↓ ↓H␈ε"zeros␈α
as␈αones,␈αwhile␈αthe␈α
subsequence␈αof␈αodd-n␈α␈um␈α␈bered␈αelemen␈α␈ts␈ε6␈α
h␈↓ 	7␈ε(X␈↓ 
%␈ε6i␈ε"␈α
=␈α
0,␈α1,
␈βs␈↓ 	U␈ε%2␈ε+n␈ε%+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␈↓ βm␈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
I␈↓ ∧ε␈ε+n
␈β
h␈↓ ↓H␈ε"obvious␈α
that␈α
the␈αsubsequence␈α
of␈α
alternate␈αterms␈ε6␈α
h␈↓ πG␈ε(U␈↓ λβ␈ε6i␈ε"␈α
=␈↓ λG␈ε(U␈↓ λq␈ε",␈↓ 	¬␈ε(U␈↓ 	/␈ε",␈↓ 	D␈ε(U␈↓ 	m␈ε",␈↓ 
α␈ε(U␈↓ 
+␈ε",␈↓ 
@␈ε".␈αε.␈αε.␈↓ 
t␈ε"will
␈β
u␈↓ π`␈ε%2␈ε+n␈↓ λ`␈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␈↓ 	6␈ε(U␈↓ 	r␈ε6i␈ε"␈α
is,␈α∞in␈α
fact,
␈β∞ ␈↓ 	N␈ε%2␈ε+␈α↓n
␈β∞?␈↓ ↓H␈ε61␈ε"-distributed,␈αand␈αm␈α␈uch␈αm␈α↓ore␈αis␈αtrue.
␈β∂↓␈↓ ↓H␈ε2De|nition␈αE.␈ε/␈α→A␈↓ βO␈ε"[␈↓ β←␈ε"0,␈αε1)␈ε/␈αsequence␈ε6␈αh␈↓ ¬K␈ε(U␈↓ ¬x␈ε6i␈ε/␈αis␈αsaid␈αto␈αbe␈ε"␈α(␈ε(m␈ε",␈↓ λ⊗␈ε(k␈↓ λ*␈ε")␈ε/-distributed␈αif
␈β∂∞␈↓ ¬d␈ε+n
␈β∂U␈↓ ↓l␈ε"P␈↓ α∧␈ε"r␈↓ α∩␈ε"(␈↓ α≡␈ε(u␈↓ αM␈ε6∀␈↓ α{␈ε(U␈↓ βx␈ε"<␈↓ ∧&␈ε(v␈↓ ∧I␈ε",␈↓ ∧←␈ε(u␈↓ ¬∞␈ε6∀␈↓ ¬<␈ε(U␈↓ εf␈ε"<␈↓ π∀␈ε(v␈↓ π6␈ε",␈↓ πL␈ε".␈αε.␈αε.␈↓ π|␈ε",␈↓ λ∩␈ε(u␈↓ λC␈ε6∀␈↓ λq␈ε(U␈↓ 
I␈ε"<␈↓ 
w␈ε(v␈↓ ~␈ε")
␈β∂a␈↓ α3␈ε%1␈↓ β∀␈ε+mn␈ε%␈α␈+␈↓ β←␈ε+j␈↓ ∧8␈ε%1␈↓ ∧s␈ε%2␈↓ ¬U␈ε+m␈α␈n␈ε%+␈↓ ε ␈ε+j␈↓ ε-␈ε%+1␈↓ π%␈ε%2␈↓ λ'␈ε+k␈↓ 	
␈ε+m␈α␈n␈ε%+␈↓ 	T␈ε+j␈↓ 	b␈ε%+␈↓ 
␈ε+k␈↓ 
⊂␈ε9␈␈ε%1␈↓ λ␈ε+k
␈β⊂⊗␈↓ πz␈ε"=␈α
(␈↓ λ4␈ε(v␈↓ λ↑␈ε6␈␈↓ 	
␈ε(u␈↓ 	0␈ε")␈↓ 	B␈ε".␈αε.␈αε.␈↓ 	r␈ε"(␈↓ 	}␈ε(v␈↓ 
*␈ε6␈␈↓ 
V␈ε(u␈↓ 
|␈ε")
␈β⊂"␈↓ λE␈ε%1␈↓ 	∨␈ε%1␈↓ 
∂␈ε+k␈↓ 
j␈ε+k
␈β⊂m␈↓ ↓H␈ε/for␈αall␈αch␈α↓oices␈αof␈αreal␈αn␈α␈um␈α␈bers␈↓ ¬*␈ε(u␈↓ ¬N␈ε/,␈↓ ¬d␈ε(v␈↓ ε⊂␈ε/with␈ε"␈α0␈ε6␈α
∀␈↓ π,␈ε(u␈↓ πZ␈ε"<␈↓ λλ␈ε(v␈↓ λ3␈ε6∀␈ε"␈α
1␈ε/␈αfor␈ε"␈α1␈ε6␈α
∀␈↓ 
␈ε(r␈↓ 
≠␈ε6∀␈↓ 
I␈ε(k␈↓ 
]␈ε/,␈αand
␈β⊂y␈↓ ¬?␈ε+r␈↓ ¬u␈ε+r␈↓ πA␈ε+r␈↓ λ~␈ε+r
␈β⊃_␈↓ ↓H␈ε/for␈αall␈αin␈α␈tegers␈↓ β7␈ε(j␈↓ βT␈ε/with␈ε"␈α0␈ε6␈α
∀␈↓ ∧p␈ε(j␈↓ ¬␈ε"<␈ε(␈α
m␈ε/.
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"149
␈β↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈βα$␈↓ ↓H␈ε"Th␈α␈us␈α
a␈↓ α=␈ε(k␈↓ αR␈ε"-distributed␈α
sequence␈α
is␈αthe␈α
special␈αcase␈ε(␈α
m␈ε"␈α
=␈α
1␈α
in␈αDe|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␈↓ ↓Z␈ε/An␈ε"␈α(␈ε(m␈ε",␈↓ αP␈ε(k␈↓ αd␈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␈↓ 	p␈ε/.␈↓ 
p␈ε"(12)
␈β∧;␈↓ ↓Z␈ε/An␈ε"␈α(␈ε(m␈ε",␈↓ αP␈ε(k␈↓ αd␈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"W␈α⎇e␈α⊂can␈α⊂also␈α⊃de|ne␈α⊂the␈α⊃concept␈α⊂of␈α⊂an␈α⊃(␈ε(m␈ε"␈α␈,␈↓ εs␈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␈↓ βg␈ε"Niv␈α␈en␈α∞and␈α∞H.␈α∞S.␈↓ ¬w␈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␈αλwill␈α	pro␈α␈v␈α␈e␈αλthat␈ε/␈α	an␈α␈y␈ε6␈α	1␈ε/-distributed␈ε(␈αλb␈ε/-ary␈α	sequence␈↓ λb␈ε(X␈↓ 	⊂␈ε/,␈↓ 	$␈ε(X␈↓ 	R␈ε/,␈↓ 	f␈ε".␈αε.␈αε.␈↓ 
_␈ε/is␈ε"␈α	(␈ε(m␈ε",␈ε(␈αεm␈ε"␈α␈)␈ε/-
␈β	N␈↓ 	␈ε%0␈↓ 	B␈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),␈↓ 
$␈ε"103↑109␈↓ "␈ε".
␈β
D␈↓ α␈ε"The␈α∞k␈α␈ey␈α∞idea␈α∞w␈α␈e␈α∞shall␈α
use␈α∞is␈α∞an␈α∞importan␈α␈t␈α∞technique␈α∞that␈α∞applies␈α∞to␈α
man␈α␈y
␈β
o␈↓ ↓H␈ε"situations␈α⊂in␈α⊂mathematics:␈α∀\If␈α⊂the␈α⊃sum␈α⊂of␈ε(␈α⊂m␈ε"␈α⊂quan␈α␈tities␈α⊂and␈α⊂the␈α⊂sum␈α⊂of␈α⊂their
␈β~␈↓ ↓H␈ε"squares␈αare␈αboth␈αconsisten␈α␈t␈αwith␈αthe␈αh␈α␈ypothesis␈αthat␈αthe␈ε(␈αm␈ε"␈αquan␈α␈tities␈αare␈αequal,
␈βE␈↓ ↓H␈ε"then␈αthat␈α
h␈α␈ypothesis␈αis␈α
true."␈α⊃In␈α
a␈α
strong␈αform,␈α
this␈αprinciple␈α
ma␈α␈y␈αbe␈α
stated␈αas
␈βp␈↓ ↓H␈ε"follo␈α␈ws:
␈β1␈↓ ↓H␈ε2Lemma␈αE.␈ε/␈α→Giv␈α␈en␈ε(␈αm␈ε/␈αsequences␈αof␈αn␈α␈um␈α␈bers␈ε6␈αh␈↓ ε␈␈ε(y␈↓ π2␈ε6i␈ε"␈α
=␈↓ πv␈ε(y␈↓ λ&␈ε/,␈↓ λ<␈ε(y␈↓ λl␈ε/,␈↓ 	α␈ε".␈αε.␈αε.␈↓ 	8␈ε/for␈ε"␈α1␈ε6␈α
∀␈↓ 
:␈ε(j␈↓ 
U␈ε6∀␈ε(␈α
m␈ε/␈α␈,
␈β>␈↓ π⊃␈ε+j␈↓ π≡␈ε+n␈↓ λλ␈ε+j␈↓ λ⊗␈ε%0␈↓ λN␈ε+j␈↓ λ[␈ε%1
␈β\␈↓ ↓H␈ε/suppose␈αthat
␈β
α␈↓ ∧-␈ε"lim␈↓ ∧m␈ε"(␈↓ ∧y␈ε(y␈↓ ¬6␈ε"+␈↓ ¬b␈ε(y␈↓ ε∨␈ε"+␈↓ εK␈ε6↓␈αε↓␈αε↓␈↓ ε⎇␈ε"+␈↓ π)␈ε(y␈↓ πi␈ε")␈↓ π␈␈ε"=␈ε(␈α
m␈ε",
␈β
∞␈↓ ¬␈ε%1␈ε+n␈↓ ¬t␈ε%2␈ε+n␈↓ π;␈ε+m␈α␈n
␈β
"␈↓ ∧≡␈ε+n␈ε9␈α␈!1
␈β
3␈↓ 
p␈ε"(14)
␈β
A␈↓ ¬λ␈ε%2␈↓ ¬r␈ε%2␈↓ π;␈ε%2␈↓ λd␈ε%2
␈β
I␈↓ βu␈ε"lim␈αεsup␈↓ ∧i␈ε"(␈↓ ∧u␈ε(y␈↓ ¬3␈ε"+␈↓ ¬←␈ε(y␈↓ ε≡␈ε"+␈↓ εJ␈ε6↓␈αε↓␈αε↓␈↓ ε|␈ε"+␈↓ π(␈ε(y␈↓ πi␈ε")␈↓ π␈␈ε6∀␈ε(␈α
m␈↓ λM␈ε(␈↓ λu␈ε".
␈β
Y␈↓ ¬λ␈ε%1␈ε+n␈↓ ¬r␈ε%2␈ε+␈α↓n␈↓ π;␈ε+m␈α␈n
␈β
p␈↓ ∧∧␈ε+n␈ε9!1
␈β∞"␈↓ ↓H␈ε/Then␈αfor␈αeach␈↓ β0␈ε(j␈↓ βA␈ε/,␈↓ βW␈ε"lim␈↓ ∧↑␈ε(y␈↓ ¬≥␈ε/exists␈αand␈αequals␈ε(␈α␈ε/.
␈β∞/␈↓ ∧	␈ε+n␈ε9␈α␈!1␈↓ ∧p␈ε+j␈↓ ∧⎇␈ε+n
␈β∞c␈↓ ↓H␈ε"An␈αincredibly␈αsimple␈αpro␈α↓of␈αof␈αthis␈αlemma␈αis␈αgiv␈α␈en␈αin␈αexercise␈α9.
␈β∞l␈↓ 	Q␈∧∞l	Q≠∂
␈β∂$␈↓ α␈ε"Resuming␈αour␈α
pro␈α↓of␈αof␈αTheorem␈αC␈↓ ε0␈ε",␈αlet␈ε(␈α
x␈ε"␈α
=␈↓ πG␈ε(x␈↓ πl␈ε(x␈↓ λ↔␈ε".␈αε.␈αε.␈↓ λG␈ε(x␈↓ 	∧␈ε"be␈αa␈ε(␈αb␈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␈αλthat␈ε(␈α	x␈ε/␈α	occurs␈ε"␈α	at␈α	position␈ε(␈α	p␈ε"␈α	of␈αλthe␈α	sequence␈α	if␈↓ πr␈ε(X␈↓ 	ε␈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␈↓ αQ␈ε+j
␈β⊂%␈↓ ↓H␈ε(p␈↓ ↓`␈ε"mod␈↓ α*␈ε(m␈ε"␈α	=␈↓ β↓␈ε(j␈↓ β∩␈ε".␈α⊂Let␈↓ βn␈ε(y␈↓ ∧,␈ε"=␈↓ ∧Z␈ε(↔␈↓ ∧|␈ε"(␈ε(n␈ε"␈α␈)/␈ε(n␈ε";␈αw␈α␈e␈αwish␈αto␈αpro␈α␈v␈α␈e␈αthat
␈β⊂2␈↓ ∧␈ε+j␈↓ ∧∞␈ε+n␈↓ ∧m␈ε+j
␈β⊂s␈↓ π1␈ε+m
␈β⊂{␈↓ ¬,␈ε"lim␈↓ ¬r␈ε(y␈↓ ε0␈ε"=␈α
1/␈ε(m␈↓ π"␈ε(b␈↓ πM␈ε".␈↓ 
p␈ε"(15)
␈β⊃λ␈↓ ε∧␈ε+j␈↓ ε∩␈ε+n
␈β⊃≠␈↓ ¬≥␈ε+n␈ε9!␈α␈1
␈β∪(

␈β↓U␈↓ ↓H␈ε"150␈↓ 
}␈ε"3.5
␈β↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈βα$␈↓ α␈ε"First␈αw␈α␈e␈αkn␈α↓o␈α␈w␈αthat
␈βαz␈↓ λi␈ε+m
␈ββ↓␈↓ βs␈ε"lim␈↓ ∧4␈ε"(␈↓ ∧@␈ε(y␈↓ ∧⎇␈ε"+␈↓ ¬)␈ε(y␈↓ ¬f␈ε"+␈↓ ε∩␈ε6↓␈αε↓␈αε↓␈↓ εD␈ε"+␈↓ εp␈ε(y␈↓ πr␈ε")␈α
=␈α
1/␈↓ λZ␈ε(b␈↓ 	¬␈ε",␈↓ 
p␈ε"(16)
␈ββ∞␈↓ ∧R␈ε%0␈ε+n␈↓ ¬;␈ε%1␈ε+n␈↓ π↓␈ε%(␈ε+␈α↓m␈ε9␈α␈␈␈ε%1)␈ε+␈α↓n
␈ββ"␈↓ βe␈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␈↓ ∧L␈ε%2␈↓ ¬6␈ε%2␈↓ ε}␈ε%2␈↓ 	ε␈ε%2␈ε+m
␈β∧x␈↓ β9␈ε"lim␈αεsup␈↓ ∧-␈ε"(␈↓ ∧9␈ε(y␈↓ ∧w␈ε"+␈↓ ¬#␈ε(y␈↓ ¬a␈ε"+␈↓ ε
␈ε6↓␈αε↓␈αε↓␈↓ ε?␈ε"+␈↓ εk␈ε(y␈↓ πo␈ε")␈ε6␈α
∀␈ε"␈α
1/␈ε(m␈↓ λw␈ε(b␈↓ 	1␈ε".␈↓ 
p␈ε"(17)
␈β¬λ␈↓ ∧L␈ε%0␈ε+n␈↓ ¬6␈ε%1␈ε+n␈↓ ε}␈ε%(␈ε+␈α↓m␈ε9␈α␈␈␈ε%1)␈ε+␈α↓n
␈β¬∨␈↓ βH␈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␈↓ ¬P␈ε%<␈ε+m
␈βλ
␈↓ ↓H␈ε"This␈αis␈αthe␈αn␈α␈um␈α␈ber␈αof␈αpairs␈αof␈αoccurrences␈αof␈ε(␈αx␈ε"␈αin␈αpositions␈↓ λp␈ε(p␈↓ 	∪␈ε",␈↓ 	#␈ε(p␈↓ 	Q␈ε"with␈ε(␈αn␈ε6␈απ␈␈↓ 
k␈ε(q␈↓ λ␈ε6∀
␈βλ⊗␈↓ 	α␈ε%1␈↓ 	5␈ε%2
␈βλ5␈↓ ↓H␈ε(p␈↓ ↓t␈ε"<␈↓ α"␈ε(p␈↓ αO␈ε"<␈ε(␈α
n␈ε"␈αand␈αwith␈↓ ∧7␈ε(p␈↓ ∧b␈ε6␈␈↓ ¬∞␈ε(p␈↓ ¬=␈ε"a␈αm␈α␈ultiple␈αof␈ε(␈αm␈ε"␈α␈.␈α⊂Consider␈αn␈α↓o␈α␈w␈αthe␈αsum
␈βλA␈↓ ↓Z␈ε%1␈↓ α4␈ε%2␈↓ ∧I␈ε%2␈↓ ¬ ␈ε%1
␈βλp␈↓ ε(␈ε↓X
␈β	∩␈↓ ¬∧␈ε(S␈↓ ¬@␈ε"=␈↓ π≤␈ε(C␈↓ π9␈ε"(␈ε(n␈ε"␈α␈).␈↓ 
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␈αof␈α
occurrences␈αof␈ε(␈α
x␈ε"␈αin␈αpositions␈↓ εc␈ε(p␈↓ πε␈ε",␈↓ π⊗␈ε(p␈↓ πE␈ε"with␈↓ λ↔␈ε(p␈↓ λE␈ε"<␈↓ λs␈ε(p␈↓ 	!␈ε"<␈↓ 	O␈ε(p␈↓ 	{␈ε"+␈↓ 
'␈ε(q␈↓ 
:␈ε",␈αwhere
␈β
∨␈↓ εu␈ε%1␈↓ π(␈ε%2␈↓ λ)␈ε%1␈↓ 	¬␈ε%2␈↓ 	a␈ε%1
␈β
>␈↓ ↓H␈ε(p␈↓ ↓t␈ε6␈␈↓ α!␈ε(p␈↓ αR␈ε"is␈α∂a␈α∞m␈α␈ultiple␈α∞of␈ε(␈α∞m␈ε"␈α∞and␈↓ ¬J␈ε(p␈↓ ¬{␈ε6∀␈↓ ε,␈ε(N␈↓ εO␈ε",␈α∂is␈α∞coun␈α␈ted␈↓ λ↔␈ε(p␈↓ λD␈ε"+␈↓ λq␈ε(q␈↓ 	∞␈ε6␈␈↓ 	;␈ε(p␈↓ 	l␈ε"times␈α∞in␈α∞the
␈β
K␈↓ ↓Z␈ε%2␈↓ α3␈ε%1␈↓ ¬\␈ε%1␈↓ λ)␈ε%1␈↓ 	M␈ε%2
␈β
i␈↓ ↓H␈ε"total␈↓ α∨␈ε(S␈↓ α`␈ε"(namely,␈α∞when␈↓ ∧T␈ε(p␈↓ ¬∧␈ε"<␈ε(␈α
n␈ε6␈α∀␈↓ ελ␈ε(p␈↓ ε4␈ε"+␈↓ εa␈ε(q␈↓ εt␈ε");␈α∞and␈α∞the␈α∞pairs␈α
of␈α∞such␈α
occurrences
␈β
v␈↓ α5␈ε+N␈↓ ∧f␈ε%2␈↓ ε~␈ε%1
␈β∃␈↓ ↓H␈ε"with␈↓ α~␈ε(N␈↓ αG␈ε"<␈↓ αu␈ε(p␈↓ β!␈ε"<␈↓ βO␈ε(p␈↓ β|␈ε"<␈↓ ∧*␈ε(N␈↓ ∧U␈ε"+␈↓ ¬↓␈ε(q␈↓ ¬ ␈ε"are␈αcoun␈α␈ted␈↓ εc␈ε(N␈↓ π∞␈ε"+␈↓ π:␈ε(q␈↓ πU␈ε6␈␈↓ λα␈ε(p␈↓ λ0␈ε"times.
␈β!␈↓ βπ␈ε%1␈↓ βa␈ε%2␈↓ λ∀␈ε%2
␈βB␈↓ α␈ε"Let␈↓ αM␈ε(d␈↓ αm␈ε"(␈ε(n␈ε"␈α␈)␈αbe␈αthe␈αn␈α␈um␈α␈ber␈αof␈αpairs␈αof␈αoccurrences␈αof␈ε(␈αx␈ε"␈αin␈αpositions␈↓ 
¬␈ε(p␈↓ 
(␈ε",␈↓ 
8␈ε(p␈↓ 
f␈ε"with
␈βO␈↓ α`␈ε+t␈↓ 
↔␈ε%1␈↓ 
J␈ε%2
␈βm␈↓ ↓H␈ε(p␈↓ ↓r␈ε"+␈ε(␈αλt␈ε"␈α=␈↓ αd␈ε(p␈↓ β⊃␈ε"<␈ε(␈α
n␈ε"␈α␈.␈α⊂The␈αanalysis␈αabo␈α␈v␈α␈e␈αsh␈α↓o␈α␈ws␈αthat
␈βz␈↓ ↓Z␈ε%1␈↓ αv␈ε%2
␈β)␈↓ ↓v␈ε↓X␈↓ εr␈ε↓X
␈βK␈↓ αY␈ε"(␈↓ αe␈ε(q␈↓ β␈ε6␈␈ε(␈αλmt␈ε")␈↓ βe␈ε(d␈↓ ∧∨␈ε"(␈↓ ∧+␈ε(N␈↓ ∧V␈ε"+␈↓ ¬α␈ε(q␈↓ ¬∃␈ε")␈ε6␈α
∃␈↓ ¬Y␈ε(S␈↓ ε⊗␈ε6∃␈↓ πU␈ε"(␈↓ πa␈ε(q␈↓ π|␈ε6␈␈ε(␈αλmt␈ε")␈↓ λa␈ε(d␈↓ 	≠␈ε"(␈↓ 	'␈ε(N␈↓ 	J␈ε").␈↓ 
p␈ε"(20)
␈βW␈↓ βw␈ε+mt␈↓ ¬n␈ε+N␈↓ λs␈ε+mt
␈β{␈↓ ↓H␈ε%0<␈ε+␈α␈t␈ε%␈α↓<␈↓ α≥␈ε+q␈↓ α-␈ε%/␈ε+␈α↓m␈↓ εD␈ε%0<␈ε+␈α␈t␈ε%␈α↓<␈↓ π→␈ε+q␈↓ π)␈ε%/␈ε+␈α↓m
␈β
M␈↓ ↓H␈ε"Since␈αthe␈αoriginal␈αsequence␈αis␈↓ ¬≡␈ε(q␈↓ ¬1␈ε"-distributed,
␈β∞!␈↓ ¬[␈ε"1
␈β∞1␈↓ πY␈ε%2␈ε+m
␈β∞9␈↓ ¬β␈ε"lim␈↓ ¬y␈ε(d␈↓ ε3␈ε"(␈↓ ε?␈ε(N␈↓ εb␈ε")␈α
=␈α
1/␈↓ πJ␈ε(b␈↓ 
p␈ε"(21)
␈β∞E␈↓ ε␈ε+mt
␈β∞M␈↓ ¬R␈∧∞M¬Rα#
␈β∞R␈↓ ¬R␈ε(N
␈β∞Y␈↓ ∧p␈ε+N␈↓ ¬␈ε9!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␈↓ ∧s␈ε↓X
␈β∂}␈↓ βV␈ε(S
␈β⊂
␈↓ βk␈ε+N
␈β⊂∞␈↓ πβ␈ε%2␈ε+m␈↓ 	K␈ε%2␈ε+␈α↓m
␈β⊂⊗␈↓ βε␈ε"lim␈↓ ∧⊗␈ε"=␈↓ ¬V␈ε"(␈↓ ¬b␈ε(q␈↓ ¬⎇␈ε6␈␈ε(␈αλmt␈ε")/␈↓ εt␈ε(b␈↓ π8␈ε"=␈↓ πf␈ε(q␈↓ πz␈ε"(␈↓ λε␈ε(q␈↓ λ!␈ε6␈␈ε(␈αλm␈ε"␈α␈)/2␈ε(m␈↓ 	<␈ε(b␈↓ 	w␈ε".␈↓ 
p␈ε"(22)
␈β⊂*␈↓ βV␈∧⊂*βVα3
␈β⊂0␈↓ β↑␈ε(N
␈β⊂6␈↓ αs␈ε+N␈↓ β∂␈ε9!1
␈β⊂F␈↓ ∧D␈ε%0␈α↓<␈ε+␈α␈t␈ε%␈α↓<␈↓ ¬~␈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↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈βα$␈↓ α␈ε"By␈αde|nition,
␈βαF␈↓ β3␈ε↓X␈↓ ∧S␈ε↓X
␈βαI␈↓ ¬(␈ε↓␈␈↓ 
←␈ε↓↓
␈βα`␈↓ πh␈ε%2
␈βαh␈↓ ↓⎇␈ε"2␈↓ α∂␈ε(S␈↓ αK␈ε"=␈↓ ¬6␈ε"(␈↓ ¬B␈ε(↔␈↓ ¬c␈ε"(␈ε(n␈ε")␈ε6␈αλ␈␈↓ εE␈ε(↔␈↓ εg␈ε"(␈ε(n␈ε6␈αλ␈␈↓ π=␈ε(q␈↓ πP␈ε")␈↓ π\␈ε")␈↓ λ↓␈ε6␈␈ε"␈αλ(␈↓ λ9␈ε(↔␈↓ λ[␈ε"(␈ε(n␈ε")␈ε6␈αλ␈␈↓ 	=␈ε(↔␈↓ 	↑␈ε"(␈ε(n␈ε6␈αλ␈␈↓ 
4␈ε(q␈↓ 
G␈ε"))␈↓ 
m␈ε",
␈βαt␈↓ α$␈ε+N␈↓ ¬T␈ε+j␈↓ εX␈ε+j␈↓ λL␈ε+j␈↓ 	O␈ε+j
␈ββ_␈↓ αy␈ε%1␈ε9␈α↓∀␈ε+␈α␈n␈ε9∀␈↓ βV␈ε+N␈↓ βr␈ε%+␈↓ ∧⊂␈ε+q␈↓ ∧3␈ε%0␈ε9␈α↓∀␈↓ ∧`␈ε+j␈↓ ∧n␈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␈↓ λC␈ε+m
␈β∧'␈↓ ∧∨␈ε"lim␈↓ ¬.␈ε"=␈↓ ¬\␈ε(q␈↓ ¬o␈ε"(␈↓ ¬{␈ε(q␈↓ ε⊗␈ε6␈␈ε(␈αλm␈ε")/␈ε(m␈↓ π ␈ε(b␈↓ πb␈ε"+␈↓ λ∞␈ε(q␈↓ λ!␈ε"/␈↓ λ3␈ε(b␈↓ λ←␈ε",␈↓ 
p␈ε"(23)
␈β∧;␈↓ ∧n␈∧∧;∧nα3
␈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␈ε↓↓
␈β¬⊗␈↓ 	∀␈ε%2
␈β¬$␈↓ βE␈ε(T␈↓ ∧↓␈ε"=␈↓ εk␈ε(↔␈↓ π
␈ε"(␈ε(n␈ε")␈ε6␈αλ␈␈↓ πo␈ε(↔␈↓ λ⊃␈ε"(␈ε(n␈ε6␈αλ␈␈↓ λg␈ε(q␈↓ λz␈ε")␈↓ 	%␈ε".
␈β¬1␈↓ βZ␈ε+N␈↓ ε}␈ε+j␈↓ λα␈ε+j
␈β¬U␈↓ ∧/␈ε%1␈ε9␈α↓∀␈ε+␈α␈n␈ε9∀␈↓ ¬␈ε+N␈↓ ¬(␈ε%+␈↓ ¬F␈ε+q␈↓ ¬i␈ε%0␈ε9∀␈↓ ε⊗␈ε+j␈↓ ε$␈ε%<␈ε+␈α␈m
␈βε∧␈↓ α␈ε"Using␈αthe␈αinequalit␈α␈y
␈βε>␈↓ ε8␈ε%2
␈βε@␈↓ ¬,␈ε↓X␈↓ π~␈ε↓X
␈βεI␈↓ ∧|␈ε↓∩␈↓ ε"␈ε↓∪
␈βεK␈↓ ∧f␈ε"1
␈βε[␈↓ λ↓␈ε%2
␈βεc␈↓ ε␈ε(a␈↓ εS␈ε6∀␈↓ πn␈ε(a
␈βεo␈↓ ε∪␈ε+j
␈βεs␈↓ λ↓␈ε+j
␈βεw␈↓ ∧f␈∧εw∧fα∩
␈βε|␈↓ ∧g␈ε(r
␈βπ∪␈↓ ¬∩␈ε%1␈ε9∀␈↓ ¬?␈ε+j␈↓ ¬M␈ε9∀␈↓ ¬j␈ε+r␈↓ π↓␈ε%1␈ε9∀␈↓ π-␈ε+j␈↓ π;␈ε9∀␈↓ πY␈ε+r
␈βπL␈↓ ↓H␈ε"(cf.␈αexercise␈α1.2.3↑30),␈αw␈α␈e␈α|nd␈αthat
␈βλλ␈↓ 	2␈ε%2
␈βλ␈↓ β ␈ε↓X␈↓ ¬w␈ε↓X
␈βλ∞␈↓ εe␈ε↓␈␈↓ 	∞␈ε↓↓
␈βλ∀␈↓ ¬(␈ε↓∩␈↓ 	≤␈ε↓∪
␈βλ∃␈↓ ∧H␈ε"1
␈βλ-␈↓ α␈ε"lim␈αεsup␈↓ εs␈ε(↔␈↓ π∃␈ε"(␈ε(n␈ε")␈ε6␈αλ␈␈↓ πw␈ε(↔␈↓ λ→␈ε"(␈ε(n␈ε6␈αλ␈␈↓ λo␈ε(q␈↓ 	α␈ε")
␈βλ9␈↓ πε␈ε+j␈↓ λ
␈ε+j
␈βλA␈↓ β}␈∧λAβ}α↓&
␈βλF␈↓ β}␈ε(N␈↓ ∧!␈ε"(␈↓ ∧-␈ε(N␈↓ ∧X␈ε"+␈↓ ¬∧␈ε(q␈↓ ¬_␈ε")
␈βλT␈↓ α↔␈ε+N␈↓ α3␈ε9!␈α␈1
␈βλ]␈↓ β␈ε%0␈ε9∀␈↓ β-␈ε+j␈↓ β;␈ε%<␈ε+␈α␈m␈↓ ¬>␈ε%1␈ε9∀␈ε+␈α␈n␈ε9∀␈↓ ε≠␈ε+N␈↓ ε6␈ε%+␈↓ εT␈ε+q
␈β	↓␈↓ λx␈ε%2␈ε+m␈↓ 
␈ε+m
␈β		␈↓ εw␈ε6∀␈↓ π%␈ε(q␈↓ π8␈ε"(␈↓ πD␈ε(q␈↓ π←␈ε6␈␈ε(␈αλm␈ε")/␈ε(m␈↓ λi␈ε(b␈↓ 	+␈ε"+␈↓ 	W␈ε(q␈↓ 	j␈ε"/␈↓ 	|␈ε(b␈↓ 
(␈ε".␈↓ 
p␈ε"(24)
␈β	L␈↓ ↓H␈ε"W␈α⎇e␈αalso␈αha␈α␈v␈α␈e
␈β	n␈↓ βA␈ε↓X␈↓ ¬⎇␈ε↓X
␈β	r␈↓ εk␈ε↓␈␈↓ 	∀␈ε↓↓
␈β
⊂␈↓ ↓S␈ε(q␈↓ ↓l␈ε(↔␈↓ α∞␈ε"(␈↓ α~␈ε(N␈↓ α=␈ε")␈ε6␈α
∀␈↓ ∧<␈ε(↔␈↓ ∧↑␈ε"(␈ε(n␈ε"␈α␈)␈α
=␈↓ εy␈ε(↔␈↓ π≠␈ε"(␈ε(n␈ε")␈ε6␈αλ␈␈↓ π⎇␈ε(↔␈↓ λ∨␈ε"(␈ε(n␈ε6␈αλ␈␈↓ λu␈ε(q␈↓ 	λ␈ε")␈↓ 	,␈ε6∀␈↓ 	Z␈ε(q␈↓ 	s␈ε(↔␈↓ 
∃␈ε"(␈↓ 
!␈ε(N␈↓ 
L␈ε"+␈↓ 
x␈ε(q␈↓ ␈ε"),
␈β
≥␈↓ ↓␈␈ε+j␈↓ ∧N␈ε+j␈↓ π␈ε+j␈↓ λ⊂␈ε+j␈↓ 
ε␈ε+j
␈β
A␈↓ β↓␈ε+N␈↓ β≥␈ε%<␈ε+n␈ε9␈α␈∀␈↓ βk␈ε+N␈↓ ∧ε␈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&␈↓ ∧c␈ε↓␈␈↓ εβ␈ε↓↓
␈β6␈↓ ε⊃␈ε%2
␈β=␈↓ λ-␈ε%2␈ε+m␈↓ 	S␈ε+m
␈βE␈↓ α{␈ε"lim␈αεsup␈↓ ∧q␈ε(↔␈↓ ¬∪␈ε"(␈↓ ¬∨␈ε(N␈↓ ¬B␈ε")/␈↓ ¬`␈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␈↓ 	D␈ε(b␈↓ 	o␈ε".␈↓ 
p␈ε"(25)
␈βQ␈↓ ¬∧␈ε+j
␈βl␈↓ βε␈ε+N␈↓ β"␈ε9!␈α␈1
␈βu␈↓ βo␈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↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈βα$␈↓ α␈ε"The␈αexistence␈αof␈ε6␈α1␈ε"-distributed␈αsequences␈αof␈αa␈αrather␈αsimple␈αt␈α␈ype␈αis␈αguaran-
␈βαO␈↓ ↓H␈ε"teed␈αby␈αthe␈αnext␈αtheorem.
␈ββ(␈↓ ↓H␈ε2Theorem␈αF␈ε"␈α(J.␈↓ β8␈ε"F␈α⎇ranklin)␈ε2.␈ε/␈α→The␈↓ ¬6␈ε"[␈↓ ¬F␈ε"0,␈αε1)␈ε/␈αsequence␈↓ π&␈ε(U␈↓ πP␈ε/,␈↓ πf␈ε(U␈↓ λ∂␈ε/,␈↓ λ%␈ε".␈αε.␈αε.␈↓ λU␈ε/,␈αwith
␈ββ4␈↓ π?␈ε%0␈↓ π␈␈ε%1
␈β∧␈↓ ε@␈ε+n
␈β∧λ␈↓ ¬>␈ε(U␈↓ ¬{␈ε"=␈↓ ε/␈ε(∩␈↓ εZ␈ε"mod␈↓ π$␈ε"1␈↓ 
p␈ε"(26)
␈β∧∃␈↓ ¬W␈ε+n
␈β∧i␈↓ ↓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¬I␈↓ 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
␈βε*␈↓ ↓H␈ε/is␈αof␈αmeasure␈αzero.
␈βπα␈↓ ↓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
␈βπ-␈↓ ↓H␈ε"cited␈αbelo␈α␈w.
␈βπ6␈↓ β*␈∧π6β*≠∂
␈βλε␈↓ α␈ε"F␈α⎇ranklin␈α⊃has␈α⊃sh␈α↓o␈α␈wn␈α⊃that␈↓ ¬-␈ε(∩␈↓ ¬O␈ε"m␈α␈ust␈α⊃be␈α⊃a␈α⊃transcenden␈α␈tal␈α⊃n␈α␈um␈α␈ber␈α⊃for␈α⊃(26)␈α⊃to
␈βλ+␈↓ ¬`␈ε+n
␈βλ1␈↓ ↓H␈ε"be␈ε6␈α∞1␈ε"-distributed.␈α→The␈α∂po␈α␈w␈α␈ers␈α∞(␈↓ ¬J␈ε(→␈↓ ¬z␈ε"mod␈↓ εD␈ε"1)␈α∂ha␈α␈v␈α␈e␈α∂been␈α∂laboriously␈α∂computed␈α∞for
␈βλ\␈↓ ↓H␈ε(n␈ε6␈α	∀␈ε"␈α
10000,␈↓ β¬␈ε"using␈αm␈α␈ultiple-precision␈αarithmetic,␈αand␈αthe␈αm␈α↓ost␈αsigni|can␈α␈t␈α35␈αbits
␈β	π␈↓ ↓H␈ε"of␈α∞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
␈β	2␈↓ ↓H␈ε"a␈α⊂source␈α⊂of␈α⊃uniform␈α⊂deviates.␈α≥According␈α⊃to␈α⊂Theorem␈α⊂F␈↓ λP␈ε",␈α∩the␈α⊂probabilit␈α␈y␈α⊂that
␈β	X␈↓ β'␈ε+n
␈β	↑␈↓ ↓H␈ε"the␈α⊂po␈α␈w␈α␈ers␈α⊃(␈↓ β⊃␈ε(→␈↓ βA␈ε"mod␈↓ ∧␈ε"1)␈α⊃are␈ε6␈α⊃1␈ε"-distributed␈α⊂is␈α⊃equal␈α⊃to␈α⊂1;␈α∪y␈α␈et␈α⊃because␈α⊃there␈α⊂are
␈β
	␈↓ ↓H␈ε"uncoun␈α␈tably␈αman␈α␈y␈αreal␈αn␈α␈um␈α␈bers,␈αthis␈αgiv␈α␈es␈αus␈αn␈α↓o␈αinformation␈αas␈αto␈αwhether␈αthe
␈β
4␈↓ ↓H␈ε"sequence␈α∂is␈α∞really␈ε6␈α∂1␈ε"-distributed␈α∂or␈α∂n␈α↓ot.␈α→It␈α∞is␈α∂a␈α∂fairly␈α∂safe␈α∂bet␈α∂that␈α∂n␈α↓obody␈α∞in
␈β
←␈↓ ↓H␈ε"our␈α
lifetimes␈α
will␈α
ev␈α␈er␈ε/␈α
pro␈α␈v␈α␈e␈ε"␈α
that␈α
this␈α
particular␈αsequence␈α
is␈ε/␈α
n␈α↓ot␈ε6␈α
1␈ε"-distributed;
␈β
␈↓ ↓H␈ε"but␈α∩it␈α⊃migh␈α␈t␈α∩n␈α↓ot␈α∩be.␈α"Because␈α∩of␈α⊃these␈α∩considerations,␈α∀one␈α∩ma␈α␈y␈α⊃legitimately
␈β6␈↓ ↓H␈ε"w␈α␈onder␈α∞if␈α∞there␈α∂is␈α∞an␈α␈y␈ε/␈α∞explicit␈ε"␈α∂sequence␈α∞that␈α∞is␈ε6␈α∞1␈ε"-distributed;␈α⊂i.e.,␈ε/␈α∂is␈α∞there␈α∞an
␈βa␈↓ ↓H␈ε/algorithm␈α∂to␈α∂compute␈α∂real␈α∞n␈α␈um␈α␈bers␈↓ ε∞␈ε(U␈↓ εJ␈ε/for␈α∂all␈ε(␈α∂n␈ε6␈α∂∃␈ε"␈α∂0␈ε/,␈α⊂such␈α∂that␈α∞the␈α∂sequence
␈βm␈↓ ε'␈ε+n
␈β␈↓ ↓H␈ε6h␈↓ ↓T␈ε(U␈↓ α↓␈ε6i␈ε/␈αis␈ε6␈α1␈ε/-distributed?␈ε"␈α⊂The␈αansw␈α␈er␈αis␈αy␈α␈es,␈αas␈αsh␈α↓o␈α␈wn␈αfor␈αexample␈αby␈αD.␈αE.␈↓ 
G␈ε"Kn␈α␈uth
␈β_␈↓ ↓l␈ε+n
␈β7␈↓ ↓H␈ε"in␈ε/␈α∞BIT␈ε2␈α∂5␈ε"␈α∂(1965),␈α∂246↑250.␈α→The␈α∂sequence␈α∞constructed␈α∂there␈α∂consists␈α∂en␈α␈tirely␈α∞of
␈βb␈↓ ↓H␈ε"rational␈α
n␈α␈um␈α␈bers;␈α∞in␈α∞fact,␈α
each␈α∞n␈α␈um␈α␈ber␈↓ εA␈ε(U␈↓ ε|␈ε"has␈α
a␈α∞terminating␈α
represen␈α␈tation␈α
in
␈βo␈↓ εZ␈ε+n
␈β
∞␈↓ ↓H␈ε"the␈α∞binary␈α∂n␈α␈um␈α␈ber␈α∞system.␈α_An␈α↓other␈α∞construction␈α∂of␈α∞an␈α∂explicit␈ε6␈α∞1␈ε"-distributed
␈β
9␈↓ ↓H␈ε"sequence,␈α
somewhat␈α
m␈α↓ore␈α	complicated␈α
than␈α
the␈α
sequence␈α
just␈α	cited,␈αfollo␈α␈ws␈α	from
␈β
d␈↓ ↓H␈ε"Theorem␈α
W␈α
belo␈α␈w.␈α∪See␈α
also␈α
N.␈α
M.␈↓ ¬⎇␈ε"Korobo␈α␈v,␈ε/␈α
Izv.␈α
Ak␈α}ad.␈α
Nauk␈α
SSSR␈ε2␈α
20␈ε"␈α
(1956),
␈β∞∂␈↓ ↓H␈ε"649↑660.
␈β∞h␈↓ ↓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∂∪␈↓ ↓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∂i␈↓ ↓H␈ε"deal␈α∪of␈α∀evidence␈α∪that␈α∀the␈α∪follo␈α␈wing␈α∀statemen␈α␈t␈α∪is␈α∀a␈α∪v␈α}alid␈α∪form␈α␈ulation␈α∀of␈α∪the
␈β⊂∀␈↓ ↓H␈ε"in␈α␈tuitiv␈α␈e␈αidea␈αof␈αrandomness:
␈β⊂m␈↓ ↓H␈ε2De|nition␈α∪R1.␈ε/␈α%A␈↓ β␈␈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⊃_␈↓ ↓H␈ε/distributed␈αsequence.
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"153
␈β↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈βα$␈↓ ↓H␈ε"W␈α⎇e␈α
ha␈α␈v␈α␈e␈α
seen␈αthat␈α
sequences␈α
meeting␈αthis␈α
de|nition␈α
will␈α
satisfy␈αall␈α
the␈α
statistical
␈βαO␈↓ ↓H␈ε"tests␈αof␈αSection␈α3.3.2␈αand␈αman␈α␈y␈αm␈α↓ore.
␈βαz␈↓ 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
␈ββ&␈↓ ↓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
␈ββQ␈↓ ↓H␈ε(U␈↓ ↓q␈ε",␈↓ α␈ε(U␈↓ α4␈ε",␈↓ αN␈ε".␈αε.␈αε.␈↓ βπ␈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
␈ββ]␈↓ ↓`␈ε%0␈↓ α#␈ε%1
␈ββ|␈↓ ↓H␈ε"generator␈α∞is␈α∂sampled␈α∂to␈α∂giv␈α␈e␈α∂v␈α}alues␈↓ ε
␈ε(U␈↓ ε3␈ε",␈↓ εM␈ε(U␈↓ εv␈ε",␈↓ π⊂␈ε".␈αε.␈αε.␈↓ π@␈ε",␈α∂an␈α␈y␈α∂of␈α∂the␈α∂possible␈α∞sequences
␈β∧λ␈↓ ε"␈ε%0␈↓ εe␈ε%1
␈β∧'␈↓ ↓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-
␈β∧R␈↓ ↓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
␈β∧}␈↓ ↓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
␈β¬)␈↓ ↓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¬T␈↓ ↓H␈ε"W␈α⎇e␈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¬␈␈↓ ↓H␈ε"at␈αthe␈αbeginning␈αof␈αthis␈αsection)␈αin␈αthe␈αfollo␈α␈wing␈αw␈α␈a␈α␈y:
␈βε7␈↓ ↓H␈ε2De|nition␈αR2.␈ε/␈α→A␈↓ βd␈ε"[␈↓ βt␈ε"0,␈αε1)␈ε/␈αsequence␈ε6␈αh␈↓ ¬`␈ε(U␈↓ ε
␈ε6i␈ε/␈αis␈αde|ned␈αto␈αbe␈α\random"␈αif,␈αwhenev␈α␈er␈↓ ⊃␈ε(P
␈βεD␈↓ ¬x␈ε+n
␈βεb␈↓ ↓H␈ε/is␈α∞a␈α
propert␈α␈y␈α∞such␈α∞that␈↓ ∧E␈ε(P␈↓ ∧a␈ε"(␈ε6h␈↓ ∧y␈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␈↓ 
v␈ε(V␈↓  ␈ε6i
␈βεo␈↓ ¬∞␈ε+n␈↓ ␈ε+n
␈βπ∞␈↓ ↓H␈ε/of␈αindependen␈α␈t␈αsamples␈αof␈αrandom␈αv␈α}ariables␈αfrom␈αthe␈αuniform␈αdistribution,␈αthen
␈βπ9␈↓ ↓H␈ε(P␈↓ ↓c␈ε"(␈ε6h␈↓ ↓{␈ε(U␈↓ α(␈ε6i␈ε")␈ε/␈αis␈αtrue.
␈βπE␈↓ α∀␈ε+n
␈βπq␈↓ α␈ε"Is␈αit␈αperhaps␈αpossible␈αthat␈αDe|nition␈αR1␈αis␈αequiv␈α}alen␈α␈t␈αto␈αDe|nition␈αR2?␈α∂Let
␈βλ≤␈↓ ↓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
␈βλG␈↓ ↓H␈ε"criticisms␈αare␈αv␈α}alid.
␈βλr␈↓ 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
␈β	≥␈↓ ↓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
␈β	I␈↓ ↓H␈ε"elemen␈α␈ts␈αare␈αall␈αzero;␈αsh␈α↓ould␈αsuch␈αa␈αsequence␈αbe␈αconsidered␈αrandom?
␈β	t␈↓ 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
␈β
∨␈↓ ↓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
J␈↓ ↓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
u␈↓ ↓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?
␈βL␈↓ 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
␈βw␈↓ ↓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.
␈βM␈↓ 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;
␈βy␈↓ ↓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
$␈↓ ↓H␈ε"zero␈α	elemen␈α␈t␈α
is␈α
n␈α↓onrandom.␈α∂More␈α
generally,␈α
h␈α↓o␈α␈w␈α␈ev␈α␈er,␈αlet␈↓ λB␈ε(x␈↓ λq␈ε"be␈α
an␈α␈y␈α
|xed␈α	n␈α␈um␈α␈ber
␈β
0␈↓ λV␈ε%0
␈β
O␈↓ ↓H␈ε"bet␈α␈w␈α␈een␈α
zero␈α
and␈α
one,␈αand␈α
let␈↓ ¬∨␈ε(P␈↓ ¬E␈ε"be␈α
the␈α
propert␈α␈y␈α
that␈αn␈α↓o␈α
elemen␈α␈t␈α
of␈α
the␈α
sequence
␈β
z␈↓ ↓H␈ε"is␈α∂equal␈α∂to␈↓ α␈␈ε(x␈↓ β$␈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
␈β∞π␈↓ β∪␈ε%0
␈β∞%␈↓ ↓H␈ε"the␈α∞elemen␈α␈t␈↓ β∂␈ε(x␈↓ β4␈ε"!␈α↔W␈α⎇e␈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∞2␈↓ β#␈ε%0
␈β∞Q␈↓ ↓H␈ε/De|nition␈αR2.␈ε"␈α⊂(F␈α⎇or␈αif␈↓ ∧%␈ε(U␈↓ ∧N␈ε",␈↓ ∧d␈ε(U␈↓ ¬∞␈ε",␈↓ ¬$␈ε".␈αε.␈αε.␈↓ ¬Z␈ε"is␈αsuch␈αa␈αsequence,␈αtak␈α␈e␈↓ λ[␈ε(x␈↓ 	
␈ε"=␈↓ 	8␈ε(U␈↓ 	a␈ε".)
␈β∞]␈↓ ∧=␈ε%0␈↓ ∧⎇␈ε%1␈↓ λo␈ε%0␈↓ 	P␈ε%0
␈β∞|␈↓ 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⊂W␈α⎇e␈α
ha␈α␈v␈α␈e␈α
n␈α↓ot␈α
really␈α
sh␈α↓o␈α␈wn␈α
that␈α
R1
␈β∂R␈↓ ↓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∂⎇␈↓ ↓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⊂)␈↓ ↓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⊂T␈↓ ↓H␈ε"irrational;␈αperhaps␈αw␈α␈e␈αsh␈α↓ould␈αinsist␈αthat
␈β⊃⊗␈↓ ¬
␈ε"P␈↓ ¬"␈ε"r␈↓ ¬1␈ε"(␈↓ ¬=␈ε(U␈↓ ¬v␈ε"is␈αrational␈↓ π∀␈ε")␈α
=␈α
0
␈β⊃#␈↓ ¬U␈ε+n
␈β∪(

␈β↓U␈↓ ↓H␈ε"154␈↓ 
}␈ε"3.5
␈β↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈βα$␈↓ ↓H␈ε"for␈αa␈αrandom␈αsequence.
␈βαO␈↓ α␈ε"Note␈α∞that␈α∞the␈α∂de|nition␈α∞of␈α∞equidistribution␈α∞sa␈α␈ys␈α∞that␈↓ λi␈ε"P␈↓ 	↓␈ε"r␈↓ 	⊂␈ε"(␈ε(u␈ε6␈α
∀␈↓ 	p␈ε(U␈↓ 
*␈ε"<␈↓ 
\␈ε(v␈↓ 
o␈ε")␈α
=
␈βα\␈↓ 
λ␈ε+n
␈βαz␈↓ ↓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
␈ββ&␈↓ ↓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
␈ββ←␈↓ ¬6␈ε"P␈↓ ¬N␈ε"r␈↓ ¬\␈ε"(␈↓ ¬h␈ε(U␈↓ ε∨␈ε62␈↓ εA␈ε(S␈↓ εZ␈ε")␈α
=␈ε(␈α
⊗␈ε",␈↓ 
p␈ε"(27)
␈ββk␈↓ ε↓␈ε+n
␈β∧_␈↓ ↓H␈ε"for␈α∪all␈α∪random␈α∪sequences␈ε6␈α∪h␈↓ ¬␈ε(U␈↓ ¬-␈ε6i␈ε"."␈α%In␈α∪particular,␈α∀if␈↓ λ∞␈ε(S␈↓ λ:␈ε"is␈α∪the␈α∪set␈α∪of␈α∪rationals,
␈β∧%␈↓ ¬_␈ε+n
␈β∧D␈↓ ↓H␈ε"it␈α⊂has␈α⊂measure␈↓ βD␈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
␈β∧o␈↓ ↓H␈ε"this␈α∩generalized␈α∩sense.␈α"It␈α∩is␈α∩reasonable␈α∩to␈α∩expect␈α∩that␈α∩Theorem␈α∩B␈α∩could␈α∩be
␈β¬~␈↓ ↓H␈ε"extended␈α∂to␈↓ β∃␈ε"Lebesgue␈α⊂in␈α␈tegration␈α⊂instead␈α∂of␈α⊂Riemann␈α⊂in␈α␈tegration,␈α⊃if␈α∂propert␈α␈y
␈β¬E␈↓ ↓H␈ε"(27)␈αis␈αstipulated.␈α⊃Ho␈α␈w␈α␈ev␈α␈er,␈α
once␈αagain␈αw␈α␈e␈α
|nd␈αthat␈αde|nition␈α(27)␈α
is␈αto␈α↓o␈αstrict,
␈β¬p␈↓ ↓H␈ε"for␈ε/␈α∞n␈α↓o␈ε"␈α∂sequence␈α∞satis|es␈α∂that␈α∞propert␈α␈y.␈α_If␈↓ εt␈ε(U␈↓ π≥␈ε",␈↓ π6␈ε(U␈↓ π`␈ε",␈↓ πy␈ε".␈αε.␈αε.␈↓ λ2␈ε"is␈α∞an␈α␈y␈α∂sequence,␈α∂the␈α∞set
␈β¬⎇␈↓ π␈ε%0␈↓ πO␈ε%1
␈βε≤␈↓ ↓H␈ε(S␈↓ ↓m␈ε"=␈ε6␈αf␈↓ α/␈ε(U␈↓ αY␈ε",␈↓ αi␈ε(U␈↓ β∩␈ε",␈↓ β"␈ε".␈αε.␈αε.␈↓ βR␈ε6g␈ε"␈α∞is␈α
of␈α
measure␈α∞zero,␈α
y␈α␈et␈↓ εe␈ε"P␈↓ ε⎇␈ε"r␈↓ π␈ε"(␈↓ π↔␈ε(U␈↓ πP␈ε62␈↓ πt␈ε(S␈↓ λ∞␈ε")␈α=␈α1.␈α∀Th␈α␈us,␈α∞by␈α
the␈α
force
␈βε(␈↓ αH␈ε%0␈↓ βα␈ε%1␈↓ π0␈ε+n
␈βεG␈↓ ↓H␈ε"of␈αthe␈α
same␈α
argumen␈α␈t␈α
w␈α␈e␈α
used␈αto␈α
exclude␈α
rationals␈α
from␈α
random␈αsequences,␈α
w␈α␈e
␈βεr␈↓ ↓H␈ε"can␈αexclude␈αall␈αrandom␈αsequences.
␈βπ≥␈↓ 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
␈βπH␈↓ ↓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
␈βπt␈↓ ↓H␈ε"in␈α␈tuitiv␈α␈e␈αsense,␈αthe␈αin|nite␈αsubsequence
␈βλ-␈↓ ∧T␈ε(U␈↓ ∧}␈ε",␈↓ ¬⊗␈ε(U␈↓ ¬?␈ε",␈↓ ¬W␈ε(U␈↓ ε↓␈ε",␈↓ ε→␈ε(U␈↓ εB␈ε",␈↓ εZ␈ε".␈αε.␈αε.␈↓ π
␈ε",␈↓ π"␈ε(U␈↓ π↑␈ε",␈↓ πv␈ε".␈αε.␈αε.␈↓ 
p␈ε"(28)
␈βλ9␈↓ πN␈ε'2
␈βλ:␈↓ ∧m␈ε%0␈↓ ¬/␈ε%1␈↓ ¬p␈ε%4␈↓ ε2␈ε%9␈↓ π;␈ε+n
␈βλg␈↓ ↓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
␈β	∩␈↓ ↓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␈↓ 	f␈ε(U␈↓ 
/␈ε6␈ ␈ε"␈α
0␈α∞for
␈β	≥␈↓ 
∩␈ε'2
␈β	≡␈↓ 	␈␈ε+n
␈β	=␈↓ ↓H␈ε"all␈ε(␈αn␈ε"␈α␈,␈αthe␈αcoun␈α␈ts␈↓ βQ␈ε(↔␈↓ βv␈ε"(␈ε(n␈ε")␈αthat␈αappear␈αin␈αthe␈αtest␈αof␈↓ πE␈ε(k␈↓ πY␈ε"-distributivit␈α␈y␈αare␈αchanged␈αby
␈β	I␈↓ βd␈ε+k
␈β	g␈↓ αI␈ε6p
␈β	h␈↓ ↓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(↔␈↓ εD␈ε"(␈ε(n␈ε"␈α␈)/␈ε(n␈ε"␈α
remain␈α
unchanged.␈α∂De|nition␈α
R1
␈β	j␈↓ αg␈∧	jαgα⊗
␈β	u␈↓ ε1␈ε+k
␈β
∪␈↓ ↓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
r␈↓ ↓H␈ε2De|nition␈α∞R3.␈ε/␈α≡A␈↓ βo␈ε"[␈↓ β␈␈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.
␈βR␈↓ ↓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
␈β⎇␈↓ ↓H␈ε"sequence␈ε6␈αh␈↓ αh␈ε(U␈↓ β∃␈ε6i␈ε"␈αhas␈αa␈αm␈α↓on␈α↓otonic␈αsubsequence␈αwith␈↓ πY␈ε(U␈↓ λ→␈ε"<␈↓ λG␈ε(U␈↓ 	λ␈ε"<␈↓ 	6␈ε(U␈↓ 	w␈ε"<␈↓ 
%␈ε6↓␈αε↓␈αε↓␈↓ 
U␈ε".
␈β	␈↓ β␈ε+n␈↓ πq␈ε+s␈↓ λ`␈ε+s␈↓ 	O␈ε+s
␈β∪␈↓ π␈␈ε'0␈↓ λn␈ε'1␈↓ 	]␈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
␈βS␈↓ ↓H␈ε"somebody␈αwh␈α↓o␈αdoes␈αn␈α↓ot␈αlo␈α↓ok␈αat␈↓ ¬E␈ε(U␈↓ ¬⎇␈ε"before␈αdeciding␈αwhether␈αor␈αn␈α↓ot␈αit␈αis␈αto␈αbe␈αin
␈β`␈↓ ¬↑␈ε+n
␈β}␈↓ ↓H␈ε"the␈αsubsequence.␈α⊂The␈αfollo␈α␈wing␈αde|nition␈αn␈α↓o␈α␈w␈αsuggests␈αitself:
␈β
2␈↓ ↓H␈ε2De|nition␈α	R4.␈ε/␈α∂A␈↓ βV␈ε"[␈↓ βf␈ε"0,␈αε1)␈ε/␈α	sequence␈ε6␈α	h␈↓ ¬L␈ε(U␈↓ ¬y␈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
?␈↓ ¬e␈ε+n
␈β
↑␈↓ ↓H␈ε/algorithm␈α
that␈α
speci|es␈α
an␈αin|nite␈α
sequence␈α
of␈α
distinct␈α
n␈α↓onnegativ␈α␈e␈α
in␈α␈tegers␈↓ π␈ε(s
␈β
j␈↓ _␈ε+n
␈β∞	␈↓ ↓H␈ε/for␈ε(␈αn␈ε6␈α	∃␈ε"␈α
0␈ε/,␈αthe␈αsubsequence␈↓ ∧⎇␈ε(U␈↓ ¬4␈ε/,␈↓ ¬J␈ε(U␈↓ ε␈ε/,␈↓ ε⊗␈ε(U␈↓ εM␈ε/,␈↓ εc␈ε".␈αε.␈αε.␈↓ π→␈ε/corresponding␈αto␈αthis␈αalgorithm␈αis
␈β∞∃␈↓ ¬⊗␈ε+s␈↓ ¬b␈ε+s␈↓ ε/␈ε+s
␈β∞∨␈↓ ¬$␈ε'0␈↓ ¬p␈ε'1␈↓ ε=␈ε'2
␈β∞4␈↓ ↓H␈ε61␈ε/-distributed.
␈β∞h␈↓ α␈ε"The␈α⊂algorithms␈α⊃referred␈α⊂to␈α⊃in␈α⊂De|nition␈α⊃R4␈↓ πc␈ε"are␈α⊃e{ectiv␈α␈e␈α⊂procedures␈α⊂that
␈β∂∪␈↓ ↓H␈ε"compute␈↓ αa␈ε(s␈↓ βε␈ε",␈α∃giv␈α␈en␈ε(␈α∪n␈ε".␈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(→␈↓ α∧␈ε"mod␈↓ αN␈ε"1␈ε6i␈ε"␈α
will␈ε/␈α
n␈α↓ot␈ε"␈αsatisfy␈α
R4,␈αsince␈α
it␈αis␈α
either␈αn␈α↓ot␈α
equidistributed␈α
or␈αthere␈α
is␈α
an
␈β∂c␈↓ 	r␈ε+s
␈β∂i␈↓ ↓H␈ε"e{ectiv␈α␈e␈αλalgorithm␈αλthat␈αλdetermines␈↓ ¬j␈ε"an␈α	in|nite␈αλsubsequence␈↓ λU␈ε(s␈↓ 	α␈ε"with␈αλ(␈↓ 	\␈ε(→␈↓ 
⊗␈ε"mod␈↓ 
`␈ε"1)␈α
<
␈β∂m␈↓ 
␈ε'0
␈β∂v␈↓ λf␈ε+n
␈β⊂∂␈↓ ↓j␈ε+s␈↓ βT␈ε+s
␈β⊂∃␈↓ ↓H␈ε"(␈↓ ↓T␈ε(→␈↓ α∞␈ε"mod␈↓ αX␈ε"1)␈α<␈α(␈↓ β>␈ε(→␈↓ βx␈ε"mod␈↓ ∧B␈ε"1)␈α
<␈↓ ¬≥␈ε6↓␈αε↓␈αε↓␈↓ ¬M␈ε".␈α∃Similarly,␈ε/␈α∞n␈α↓o␈α
explicitly␈α∞de|ned␈α
sequence␈α
can
␈β⊂_␈↓ ↓x␈ε'1␈↓ βb␈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␈α
de|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␈↓ αI␈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↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈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␈↓ πL␈ε+n
␈βαO␈↓ ↓H␈ε"facts␈α
are␈α
kn␈α↓o␈α␈wn,␈αfor␈α
example:␈α∪(i)␈α
The␈α
sequence␈ε6␈α
h␈↓ π:␈ε(∩␈↓ πf␈ε"mod␈↓ λ0␈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+n␈ε%)
␈ββQ␈↓ ↓H␈ε"258.␈α∨(ii)␈α∂The␈α∂particular␈α∂sequence␈ε6␈α∂h␈↓ ¬{␈ε(∩␈↓ εJ␈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␈↓ π2␈ε",␈αor␈ε(␈αs␈ε"(␈ε(n␈ε"␈α␈)␈α=␈ε6␈α
b␈ε(n␈↓ 	∀␈ε"lg␈↓ 	6␈ε(n␈ε6c␈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␈↓ λQ␈ε(U␈↓ λ}␈ε6i␈ε"␈α
be␈α
a␈α
truly␈α	random
␈β¬␈↓ λj␈ε+n
␈β¬+␈↓ ↓H␈ε"sequence,␈α
and␈α
de|ne␈αthe␈α
subsequence␈ε6␈α
h␈↓ ε,␈ε(U␈↓ εf␈ε6i␈ε"␈α
by␈αthe␈α
follo␈α␈wing␈α
rules:␈↓ 	m␈ε(s␈↓ 
~␈ε"=␈α0,␈α
and
␈β¬7␈↓ εE␈ε+s␈↓ 	}␈ε%0
␈β¬A␈↓ εS␈ε-n
␈β¬V␈↓ ↓H␈ε"(for␈ε(␈αn␈ε"␈α
>␈α
0)␈↓ ββ␈ε(s␈↓ β4␈ε"is␈αthe␈αsmallest␈αin␈α␈teger␈ε6␈α∃␈ε(␈α
n␈ε"␈αfor␈αwhich␈↓ λ∧␈ε(U␈↓ λk␈ε",␈↓ 	↓␈ε(U␈↓ 	g␈ε",␈↓ 	⎇␈ε".␈αε.␈αε.␈↓ 
-␈ε",␈↓ 
C␈ε(U
␈β¬b␈↓ β∀␈ε+n␈↓ λ≥␈ε+s␈↓ λ<␈ε9␈␈ε%1␈↓ 	→␈ε+s␈↓ 	9␈ε9␈␈ε%␈α␈2␈↓ 
[␈ε+s␈↓ 
{␈ε9␈␈ε+␈α␈n
␈β¬l␈↓ λ+␈ε-n␈↓ 	'␈ε-n␈↓ 
i␈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(␈↓ πU␈ε%1␈↓ π␈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␈↓ πh␈ε".␈α!Suppose␈α∩that␈α∩\␈↓ 
␈ε(U␈↓ 
L␈ε"<␈↓ ~␈ε""
␈βε9␈↓ 
$␈ε+n
␈βε=␈↓ πU␈ε%2␈↓ π␈ε%2
␈βε@␈↓ πU␈∧ε@πUα⊂␈↓ π␈∧ε@πα⊂
␈βεW␈↓ ↓H␈ε"corresponds␈αto␈α
the␈αv␈α}alue␈α\heads"␈αin␈αthe␈α⎇ipping␈αof␈α
a␈αcoin.␈α⊂Gam␈α␈blers␈αtend␈αto␈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␈αtrue␈αcoin␈αis␈αbeing␈αused;␈αand␈αthe␈α
subsequence␈ε6␈αh␈↓ 	#␈ε(U␈↓ 	\␈ε6i␈ε"␈αjust␈αde|ned
␈βπ:␈↓ 	;␈ε+s
␈βπD␈↓ 	I␈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+␈↓ βM␈ε%1␈↓ ¬↑␈ε%1
␈βλ/␈↓ ↓H␈ε"that␈↓ α→␈ε"P␈↓ α1␈ε"r␈↓ α@␈ε"(␈↓ αL␈ε(U␈↓ β∃␈ε6∃␈↓ βa␈ε")␈α∂is␈α⊂m␈α↓ore␈α⊂than␈↓ ¬q␈ε",␈α⊃but␈α⊂of␈α∂course␈α⊂in␈α⊂a␈α∂truly␈α⊂random␈α∂sequence
␈βλ<␈↓ αd␈ε+s
␈βλ@␈↓ βM␈ε%2␈↓ ¬↑␈ε%2
␈βλD␈↓ βM␈∧λDβMα⊂␈↓ ¬↑␈∧λD¬↑α⊂
␈βλE␈↓ αr␈ε-n
␈βλ[␈↓ ↓H␈ε6h␈↓ ↓T␈ε(U␈↓ α
␈ε6i␈ε"␈α
will␈αbe␈αcompletely␈αrandom.␈α⊃No␈α
gam␈α␈bling␈αsystem␈αwill␈αev␈α␈er␈α
be␈αable␈αto␈αbeat
␈βλg␈↓ ↓l␈ε+s
␈βλq␈↓ ↓z␈ε-n
␈β	ε␈↓ ↓H␈ε"the␈α∞odds!␈α⊗De|nition␈α∞R4␈α∂sa␈α␈ys␈α∞n␈α↓othing␈α∞about␈α∞subsequences␈α∞formed␈α∞according␈α∞to
␈β	1␈↓ ↓H␈ε"such␈αa␈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␈↓ ↓z␈ε"(␈↓ αε␈ε(x␈↓ α*␈ε",␈↓ α:␈ε".␈αε.␈αε.␈↓ αj␈ε",␈↓ αz␈ε(x␈↓ β#␈ε")␈ε6i␈ε"␈α∩where,␈α∪for␈ε(␈α∩n␈ε6␈α∀∃␈ε"␈α∀0,␈↓ ε⊗␈ε(f␈↓ εN␈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␈↓ αy␈ε"(␈↓ β¬␈ε(x␈↓ β*␈ε",␈↓ β:␈ε".␈αε.␈αε.␈↓ βj␈ε",␈↓ βz␈ε(x␈↓ ∧"␈ε")␈α∞is␈α∞either␈α∂0␈α∞or␈α∞1.␈α↔Here␈↓ π"␈ε(x␈↓ πG␈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␈↓ ¬K␈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␈αsequence␈ε6␈αh␈↓ 	1␈ε(X␈↓ 	d␈ε6i␈ε"␈αof␈αelemen␈α␈ts
␈β↔␈↓ 	O␈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␈↓ ↓n␈ε"(␈↓ ↓z␈ε(X␈↓ α(␈ε",␈↓ α8␈ε(X␈↓ αg␈ε",␈↓ αw␈ε".␈αε.␈αε.␈↓ β'␈ε",␈↓ β7␈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␈↓ αV␈ε%1␈↓ βU␈ε+n␈ε9␈␈ε%␈α␈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␈↓ ¬B␈ε"=␈α
1;␈α
and␈α∞for␈ε(␈α
n␈ε"␈α>␈α0,␈↓ λ→␈ε(f␈↓ λ?␈ε"(␈↓ λK␈ε(x␈↓ λo␈ε",␈↓ λ␈␈ε".␈αε.␈αε.␈↓ 	/␈ε",␈↓ 	?␈ε(x␈↓ 	h␈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␈↓ πN␈ε%1
␈β
;␈↓ ↓H␈ε"parameters␈↓ β∧␈ε(x␈↓ β4␈ε",␈↓ βM␈ε(x␈↓ ∧*␈ε",␈↓ ∧C␈ε".␈αε.␈αε.␈↓ ∧s␈ε",␈↓ ¬␈ε(x␈↓ ε%␈ε"are␈α∂all␈α∞<␈↓ πo␈ε"when␈ε(␈α∞m␈ε"␈α∞=␈ε(␈α∞n␈ε"␈α∞but␈α∞n␈α↓ot␈α∞when
␈β
G␈↓ β_␈ε+m␈↓ βa␈ε+m␈ε9␈␈ε%␈α␈1␈↓ ¬ ␈ε+m␈ε9␈α␈␈␈↓ ¬X␈ε+k␈↓ ¬i␈ε%+␈α␈1
␈β
L␈↓ πN␈ε%2
␈β
O␈↓ πN␈∧
OπNα⊂
␈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"(␈↓ ε3␈ε(x␈↓ εX␈ε",␈↓ εh␈ε".␈αε.␈αε.␈↓ π_␈ε",␈↓ π(␈ε(x␈↓ πP␈ε"),␈α∞when␈ε(␈α∞n␈ε"␈α∞and␈↓ 	@␈ε(x␈↓ 	e␈ε",␈↓ 	}␈ε".␈αε.␈αε.␈↓ 
.␈ε",␈↓ 
F␈ε(x␈↓ 
|␈ε"are
␈β∞K␈↓ ε∪␈ε+n␈↓ εG␈ε%1␈↓ π<␈ε+n␈↓ 	T␈ε%1␈↓ 
Z␈ε+n
␈β∞j␈↓ ↓H␈ε"giv␈α␈en␈α	as␈α
input.␈α⊂W␈α⎇e␈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↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈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␈αh␈↓ ∧0␈ε(U␈↓ ∧]␈ε6i␈ε/␈α
is␈α
said␈αto␈α
be␈α
\random"␈αif␈α
the␈ε(␈α
b␈ε/-ary␈α
sequence␈ε6␈αhb␈ε(b␈↓ 
e␈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␈α
that␈αDe|nition␈α
R5␈αsa␈α␈ys␈α
only␈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␈α
ma␈α␈y␈αde|ne␈αan␈αobviously␈αcomputable␈α
subsequence␈αrule␈ε6␈αR␈ε"(␈↓ λw␈ε(a␈↓ 	!␈ε".␈αε.␈αε.␈↓ 	Q␈ε(a␈↓ 	v␈ε")␈αas␈α
follo␈α␈ws,
␈β∧S␈↓ 	
␈ε%1␈↓ 	d␈ε+k
␈β∧q␈↓ ↓H␈ε"giv␈α␈en␈α⊂an␈α␈y␈ε(␈α∂b␈ε"␈α↓-ary␈α⊂n␈α␈um␈α␈ber␈↓ ∧U␈ε(a␈↓ ∧␈␈ε".␈αε.␈αε.␈↓ ¬/␈ε(a␈↓ ¬T␈ε":␈α∀Let␈↓ ε8␈ε(f␈↓ ε↑␈ε"(␈↓ εj␈ε(x␈↓ π∂␈ε",␈↓ π∨␈ε".␈αε.␈αε.␈↓ πO␈ε",␈↓ π←␈ε(x␈↓ λπ␈ε")␈α⊂=␈α⊃1␈α⊂if␈α⊂and␈α∂only␈α⊂if␈ε(␈α⊂n␈ε6␈α⊂∃
␈β∧}␈↓ ∧h␈ε%1␈↓ ¬B␈ε+k␈↓ εJ␈ε+n␈↓ ε}␈ε%1␈↓ πs␈ε+n
␈β¬≥␈↓ ↓H␈ε(k␈↓ ↓e␈ε6␈␈ε"␈α
1␈α∞and␈↓ α|␈ε(x␈↓ ∧
␈ε"=␈↓ ∧?␈ε(a␈↓ ∧c␈ε",␈↓ ∧|␈ε".␈αε.␈αε.␈↓ ¬,␈ε",␈↓ ¬E␈ε(x␈↓ ε(␈ε"=␈↓ εZ␈ε(a␈↓ π,␈ε",␈↓ πE␈ε(x␈↓ π{␈ε"=␈↓ λ-␈ε(a␈↓ λR␈ε".␈α↔No␈α␈w␈α∂if␈ε6␈α∞h␈↓ 	y␈ε(X␈↓ 
+␈ε6i␈ε"␈α∂is␈α∞a␈↓ ␈ε(k␈↓  ␈ε"-
␈β¬)␈↓ β⊂␈ε+n␈ε9␈α␈␈␈↓ β@␈ε+k␈↓ βQ␈ε%+␈α␈1␈↓ ∧R␈ε%1␈↓ ¬Y␈ε+n␈ε9␈α␈␈␈ε%1␈↓ εm␈ε+k␈↓ ε⎇␈ε9␈␈ε%1␈↓ πY␈ε+n␈↓ λ@␈ε+k␈↓ 
↔␈ε+n
␈β¬H␈↓ ↓H␈ε"distributed␈ε(␈αb␈ε"-ary␈αsequence,␈αthis␈αrule␈ε6␈αR␈ε"(␈↓ ε+␈ε(a␈↓ εU␈ε".␈αε.␈αε.␈↓ π¬␈ε(a␈↓ π*␈ε")←which␈αselects␈αthe␈αsubsequence
␈β¬T␈↓ ε>␈ε%1␈↓ π_␈ε+k
␈β¬s␈↓ ↓H␈ε"consisting␈α∂of␈α⊂th␈α↓ose␈α∂terms␈α⊂just␈α⊂follo␈α␈wing␈α∂an␈α⊂occurrence␈α∂of␈↓ λj␈ε(a␈↓ 	∀␈ε".␈αε.␈αε.␈↓ 	D␈ε(a␈↓ 	i␈ε"←de|nes␈α∂an
␈β¬␈␈↓ λ⎇␈ε%1␈↓ 	W␈ε+k
␈βε≡␈↓ ↓H␈ε"in|nite␈α
subsequence;␈α
and␈α
if␈α
this␈α
subsequence␈α
is␈α
1-distributed,␈α
each␈α
of␈α
the␈α
(␈↓ 
B␈ε(k␈↓ 
Z␈ε"+␈α∧1)-
␈βεC␈↓ 	c␈ε+k␈↓ 	t␈ε%+␈α␈1
␈βεI␈↓ ↓H␈ε"tuples␈↓ α2␈ε(a␈↓ α\␈ε".␈αε.␈αε.␈↓ β␈ε(a␈↓ β1␈ε(a␈↓ ∧∂␈ε"for␈α0␈ε6␈α
∀␈↓ ¬⊃␈ε(a␈↓ ¬n␈ε"<␈ε(␈α
b␈ε"␈αoccurs␈αwith␈α
probabilit␈α␈y␈α1/␈↓ 	T␈ε(b␈↓ 
.␈ε"in␈ε6␈αh␈↓ 
d␈ε(X␈↓ ⊗␈ε6i␈ε".
␈βεV␈↓ αE␈ε%1␈↓ β∨␈ε+k␈↓ βD␈ε+k␈↓ βT␈ε%+1␈↓ ¬$␈ε+k␈↓ ¬5␈ε%+1␈↓ α␈ε+n
␈βεu␈↓ ↓H␈ε"Th␈α␈us␈αλw␈α␈e␈α	can␈α	pro␈α␈v␈α␈e␈α	that␈α	a␈α	sequence␈αλsatisfying␈α	De|nition␈α	R5␈α	is␈↓ λ⎇␈ε(k␈↓ 	⊃␈ε"-distributed␈α	for␈αλall
␈βπ ␈↓ ↓H␈ε(k␈↓ ↓\␈ε",␈α	by␈α
induction␈α	on␈↓ βl␈ε(k␈↓ ∧␈ε".␈α∂Similarly,␈α
by␈α	considering␈α	the␈α	\composition"␈α	of␈α	subsequence
␈βπK␈↓ ↓H␈ε"rules←if␈↓ αV␈ε6R␈↓ β
␈ε"de|nes␈α
an␈αin|nite␈α
subsequence␈ε6␈α
h␈↓ ε}␈ε(X␈↓ π1␈ε6i␈↓ π=␈ε6R␈↓ πi␈ε",␈αthen␈α
w␈α␈e␈αcan␈α
de|ne␈↓ 
)␈ε6R␈↓ 
U␈ε6R␈↓ ␈ε"to
␈βπW␈↓ αr␈ε%1␈↓ π≤␈ε+n␈↓ πY␈ε%1␈↓ 
E␈ε%1␈↓ 
q␈ε%2
␈βπv␈↓ ↓H␈ε"be␈α∞the␈α∞subsequence␈α∞rule␈α
for␈α∞which␈ε6␈α∞h␈↓ ¬␈␈ε(X␈↓ ε2␈ε6i␈↓ ε>␈ε6R␈↓ εj␈ε6R␈↓ π$␈ε"=␈α∞(␈ε6h␈↓ πn␈ε(X␈↓ λ ␈ε6i␈↓ λ,␈ε6R␈↓ λY␈ε")␈↓ λe␈ε6R␈↓ 	∩␈ε"←w␈α␈e␈α∞|nd␈α
that␈α∞all
␈βλβ␈↓ ε≥␈ε+n␈↓ εZ␈ε%1␈↓ πε␈ε%2␈↓ λ␈ε+n␈↓ λH␈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␈αthat␈α
sequences␈α
satisfying␈α
De|nition␈αR5␈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␈↓ β0␈ε(X␈↓ βo␈ε6i␈ε"␈α⊃for␈α⊂which␈↓ ¬5␈ε(s␈↓ ¬h␈ε"<␈↓ ε≡␈ε(s␈↓ εQ␈ε"<␈↓ πε␈ε6↓␈αε↓␈αε↓␈↓ π6␈ε",␈α∩but␈ε6␈α⊃h␈↓ λ%␈ε(s␈↓ λJ␈ε6i␈ε"␈α⊂does␈α⊃n␈α↓ot␈α⊂ha␈α␈v␈α␈e␈α⊃to␈α⊂be
␈β
1␈↓ βN␈ε+s␈↓ ¬F␈ε%0␈↓ ε/␈ε%1␈↓ λ6␈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≤␈↓ λT␈ε(s␈↓ 	
␈ε"for␈ε(␈αn␈ε6␈α
≤␈ε(␈α
m␈ε".
␈β
\␈↓ λλ␈ε+n␈↓ λe␈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␈↓ ¬I␈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␈↓ ¬g␈ε+n
␈βf␈↓ ↓H␈ε/algorithm␈α
that␈αspeci|es␈α
an␈αin|nite␈α
sequence␈αof␈αdistinct␈α
n␈α↓onnegativ␈α␈e␈αin␈α␈tegers␈ε6␈α
h␈↓ 
{␈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␈↓ εb␈ε/,␈↓ ε|␈ε".␈αε.␈αε.␈↓ π,␈ε/,␈↓ πG␈ε(X␈↓ λ,␈ε/,␈α⊂the␈α⊂subsequence␈ε6␈α∂h␈↓ 
a␈ε(X␈↓  ␈ε6i
␈β≡␈↓ εD␈ε+s␈↓ πe␈ε+s␈↓ 
␈␈ε+s
␈β'␈↓ εR␈ε'0␈↓ πs␈ε-n␈ε;␈␈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␈αh␈↓ ∧0␈ε(U␈↓ ∧]␈ε6i␈ε/␈α
is␈α
said␈αto␈α
be␈α
\random"␈αif␈α
the␈ε(␈α
b␈ε/-ary␈α
sequence␈ε6␈αhb␈ε(b␈↓ 
e␈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.␈α	Existence␈α	of␈α	random␈αλsequences.␈ε"␈α∂W␈α⎇e␈α	ha␈α␈v␈α␈e␈α	seen␈α	that␈α	De|nition␈α	R3␈α	is␈αλto␈α↓o␈α	strong,
␈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↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈βα$␈↓ ↓H␈ε"because␈α
w␈α␈e␈α
believ␈α␈e␈α∞that␈α
a␈α∞truly␈α
random␈α
sequence␈α∞exists␈α
and␈α∞satis|es␈α
R6;␈α∞but␈α
a
␈βαO␈↓ ↓H␈ε"pro␈α↓of␈αis␈αreally␈αnecessary␈αto␈αsh␈α↓o␈α␈w␈αthat␈αthe␈αde|nition␈αis␈αconsisten␈α␈t.
␈βαz␈↓ α␈ε"An␈α∂in␈α␈teresting␈α∞meth␈α↓od␈α∂for␈α∂constructing␈α∞sequences␈α∂satisfying␈α∂De|nition␈α∞R5
␈ββ&␈↓ ↓H␈ε"has␈αbeen␈αfound␈αby␈αA.␈↓ ∧%␈ε"W␈α⎇ald,␈αstarting␈↓ ε∩␈ε"with␈αa␈αv␈α␈ery␈αsimple␈α1-distributed␈αsequence.
␈ββa␈↓ ↓H␈ε2Lemma␈α∂T.␈ε/␈α≡Let␈α⊂the␈α∂sequence␈α∂of␈α∂real␈α∂n␈α␈um␈α␈bers␈ε6␈α∂h␈↓ πB␈ε(V␈↓ πl␈ε6i␈ε/␈α∂be␈α∂de|ned␈α∂in␈α∂terms␈α∂of␈α∂the
␈ββn␈↓ πX␈ε+n
␈β∧␈↓ ↓H␈ε/binary␈αsystem␈αas␈αfollo␈α␈ws:
␈β∧O␈↓ ↓Z␈ε(V␈↓ α
␈ε"=␈α
0,␈↓ β"␈ε(V␈↓ βS␈ε"=␈α
.1,␈↓ ∧u␈ε(V␈↓ ¬%␈ε"=␈α
.01,␈↓ εY␈ε(V␈↓ π
␈ε"=␈α
.11,␈↓ λ>␈ε(V␈↓ λn␈ε"=␈α
.001,␈↓ 
4␈ε".␈αε.␈αε.
␈β∧[␈↓ ↓o␈ε%0␈↓ β8␈ε%1␈↓ ¬␈ε%2␈↓ εo␈ε%3␈↓ λT␈ε%4
␈β¬λ␈↓ ¬⊗␈ε+r␈↓ ε␈ε+r␈↓ ε→␈ε9␈␈ε%1
␈β¬⊂␈↓ ↓Z␈ε(V␈↓ α∞␈ε"=␈α
.␈↓ αF␈ε(c␈↓ αj␈ε".␈αε.␈αε.␈↓ β~␈ε(c␈↓ β:␈ε"1␈↓ ∧∀␈ε/if␈↓ ∧6␈ε(n␈ε"␈α
=␈↓ ¬∧␈ε"2␈↓ ¬-␈ε"+␈↓ ¬Y␈ε(c␈↓ ¬z␈ε"2␈↓ εP␈ε"+␈↓ ε|␈ε6↓␈αε↓␈αε↓␈↓ π.␈ε"+␈↓ πZ␈ε(c␈↓ πx␈ε".␈↓ 
p␈ε"(29)
␈β¬≤␈↓ ↓o␈ε+n␈↓ αU␈ε+r␈↓ β*␈ε%1␈↓ ¬i␈ε%1␈↓ πi␈ε+r
␈β¬Z␈↓ ↓H␈ε/Let␈↓ απ␈ε(I␈↓ αp␈ε/den␈α↓ote␈α	the␈α	set␈α	of␈α	all␈α	real␈αλn␈α␈um␈α␈bers␈α	in␈↓ π≡␈ε"[␈↓ π/␈ε"0,␈αε1)␈ε/␈α	wh␈α↓ose␈α	binary␈αλrepresen␈α␈tation
␈β¬g␈↓ α∃␈ε+b␈↓ α1␈ε%.␈α␈..␈↓ αK␈ε+b
␈β¬p␈↓ α"␈ε'1␈↓ αX␈ε-r
␈βε¬␈↓ ↓H␈ε/begins␈αwith␈ε"␈α0.␈↓ β$␈ε(b␈↓ βJ␈ε".␈αε.␈αε.␈↓ βz␈ε(b␈↓ ∧_␈ε/;␈αth␈α␈us
␈βε∩␈↓ β3␈ε%1␈↓ ∧	␈ε+r
␈βε1␈↓ ∧v␈ε↓α␈↓ λ}␈ε↓↓
␈βεH␈↓ λQ␈ε9␈␈↓ λo␈ε+r
␈βεP␈↓ β↑␈ε(I␈↓ ∧H␈ε"=␈↓ ¬α␈ε"(0.␈↓ ¬*␈ε(b␈↓ ¬P␈ε".␈αε.␈αε.␈↓ ε␈ε(b␈↓ ε∨␈ε")␈↓ ε<␈ε",␈α(0.␈↓ εz␈ε(b␈↓ π ␈ε".␈αε.␈αε.␈↓ πP␈ε(b␈↓ πn␈ε")␈↓ λ∪␈ε"+␈↓ λ?␈ε"2␈↓ 	␈ε".␈↓ 
p␈ε"(30)
␈βε\␈↓ βm␈ε+b␈↓ ∧λ␈ε%..␈α␈.␈↓ ∧"␈ε+b␈↓ ¬:␈ε%1␈↓ ε⊂␈ε+r␈↓ ε+␈ε%2␈↓ π	␈ε%1␈↓ π←␈ε+r␈↓ πz␈ε%2
␈βεf␈↓ βy␈ε'1␈↓ ∧/␈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␈↓ ε7␈ε/in␈↓ εa␈ε(I␈↓ πN␈ε/for␈ε"␈α0␈ε6␈α
∀␈↓ λP␈ε(k␈↓ λn␈ε"<␈ε(␈α
n␈ε/,␈αw␈α␈e␈αha␈α␈v␈α␈e
␈βπ'␈↓ ε→␈ε+k␈↓ εp␈ε+b␈↓ π␈ε%.␈α␈..␈↓ π&␈ε+b
␈βπ1␈↓ ε⎇␈ε'1␈↓ π3␈ε-r
␈βπ]␈↓ εD␈ε9␈␈↓ εb␈ε+r
␈βπe␈↓ ∧⎇␈ε6j␈↓ ¬π␈ε(↔␈↓ ¬≤␈ε"(␈ε(n␈ε")/␈ε(n␈ε6␈α∞␈␈↓ ε2␈ε"2␈↓ εq␈ε6j␈α
∀␈ε"␈α
1/␈ε(n␈ε".␈↓ 
p␈ε"(31)
␈βλ*␈↓ λ5␈ε+r
␈βλ0␈↓ ↓H␈ε/Pro␈α↓of.␈ε"␈α→Since␈↓ β ␈ε(↔␈↓ β5␈ε"(␈ε(n␈ε")␈αis␈αthe␈α
n␈α␈um␈α␈ber␈αof␈↓ ¬}␈ε(k␈↓ ε≡␈ε"for␈αwhich␈↓ π?␈ε(k␈↓ πY␈ε"mod␈↓ λ#␈ε"2␈↓ λN␈ε"=␈α(␈↓ 		␈ε(b␈↓ 	-␈ε".␈αε.␈αε.␈↓ 	]␈ε(b␈↓ 	⎇␈ε")␈↓ 
~␈ε",␈αw␈α␈e␈αha␈α␈v␈α␈e
␈βλ<␈↓ 	_␈ε+r␈↓ 	l␈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"␈α␈)␈ε6␈αλ␈␈ε(␈αλn␈ε"/␈↓ λ≤␈ε"2␈↓ λ>␈ε6j␈α
∀␈ε"␈α
1.
␈βλd␈↓ 	C␈∧λd	C≠∂
␈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␈↓ πλ␈ε"2␈↓ π*␈ε(V␈↓ πT␈ε6ci␈ε"␈α
is␈α
an␈α
equidistributed␈↓ 
L␈ε"2␈↓ 
m␈ε"-ary
␈β	#␈↓ π@␈ε+n
␈β	B␈↓ ↓H␈ε"sequence;␈α	hence␈αλby␈α	Theorem␈αλA␈↓ ¬!␈ε",␈ε6␈α	h␈↓ ¬@␈ε(V␈↓ ¬j␈ε6i␈ε"␈αλis␈α	an␈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␈α
h␈↓ ∧#␈ε(V␈↓ ∧M␈ε6i␈ε"␈α
is␈αabout␈α
as␈α
equidistributed␈αas␈α
a␈↓ λQ␈ε"[␈↓ λb␈ε"0,␈αε1)␈α
sequence␈α
can␈α
be.
␈β	y␈↓ ∧9␈ε+n
␈β
_␈↓ ↓H␈ε"(F␈α⎇or␈α
further␈α
discussion␈α
of␈α
this␈αand␈α
related␈α
sequences,␈α∞see␈α
J.␈α
G.␈↓ 	0␈ε"v␈α}an␈αder␈α
Corput,
␈β
C␈↓ ↓H␈ε/Proc.␈αKoninklijk␈α␈e␈αNederl.␈αAk␈α}ad.␈αW␈α⎇etenschappen␈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",␈↓ βO␈ε6R␈↓ β|␈ε",␈↓ ∧⊂␈ε".␈αε.␈αε.␈↓ ∧C␈ε"be␈α
in|nitely␈α	man␈α␈y␈α
subsequence␈α	rules;␈αw␈α␈e␈α	seek␈α
a␈α	sequence
␈βQ␈↓ β*␈ε%1␈↓ βk␈ε%2
␈βp␈↓ ↓H␈ε6h␈↓ ↓T␈ε(U␈↓ α↓␈ε6i␈ε"␈αfor␈αwhich␈αall␈αthe␈αin|nite␈αsubsequences␈ε6␈αh␈↓ ππ␈ε(U␈↓ π4␈ε6i␈↓ π@␈ε6R␈↓ πw␈ε"are␈αequidistributed.
␈β|␈↓ ↓l␈ε+n␈↓ π∨␈ε+n␈↓ π\␈ε+j
␈β+␈↓ ↓H␈ε2Algorithm␈αW␈ε"␈α(␈ε/W␈α⎇ald␈αsequence␈↓ ¬⊗␈ε")␈ε2.␈ε"␈α~Giv␈α␈en␈αan␈αin|nite␈αsequence␈αof␈αsubsequence␈αrules
␈βW␈↓ ↓H␈ε6R␈↓ ↓t␈ε",␈↓ α	␈ε6R␈↓ α6␈ε",␈↓ αK␈ε".␈αε.␈αε.␈↓ α␈␈ε"that␈αde|ne␈α
subsequences␈αof␈↓ ε1␈ε"[␈↓ εB␈ε"0,␈αε1)␈α
sequences␈αof␈α
rational␈αn␈α␈um␈α␈bers,␈α
this
␈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␈αauxiliary␈α
v␈α}ariables␈↓ ∧X␈ε(C␈↓ ∧u␈ε"[␈↓ ∧␈␈ε(a␈↓ ¬#␈ε",␈↓ ¬3␈ε".␈αε.␈αε.␈↓ ¬c␈ε",␈↓ ¬s␈ε(a␈↓ ε∃␈ε"],␈α
where␈↓ π∨␈ε(r␈↓ π;␈ε6∃␈ε"␈α1␈α
and␈α
where␈↓ 	9␈ε(a␈↓ 	f␈ε"=␈α0␈α
or␈α
1␈αfor
␈β
9␈↓ ¬∩␈ε%1␈↓ εε␈ε+r␈↓ 	L␈ε+j
␈β
X␈↓ ↓H␈ε"1␈ε6␈α
∀␈↓ α∩␈ε(j␈↓ α-␈ε6∀␈↓ α[␈ε(r␈↓ αk␈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␈↓ β8␈ε".]␈α→Set␈↓ ∧#␈ε(r␈↓ ∧=␈ε6␈ ␈ε"␈α
1.
␈β∞r␈↓ ↓@␈ε2W3.␈↓ α	␈ε"[T␈α⎇est␈↓ α`␈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␈↓ 
1␈ε+r
␈β∂≡␈↓ α␈ε"on␈αthe␈αv␈α}alues␈αof␈↓ ∧⊃␈ε(U␈↓ ∧H␈ε"for␈α0␈ε6␈α
∀␈↓ ¬J␈ε(k␈↓ ¬h␈ε"<␈ε(␈α
n␈ε",␈αset␈↓ εz␈ε(a␈↓ π&␈ε6␈ ␈ε"␈α
1;␈αotherwise␈αset␈↓ 	T␈ε(a␈↓ 
↓␈ε6␈ ␈ε"␈α
0.
␈β∂*␈↓ ∧*␈ε+k␈↓ π
␈ε+r␈↓ 	g␈ε+r
␈β∂K␈↓ π+␈ε+r␈↓ π9␈ε9␈␈ε%1
␈β∂Q␈↓ ↓@␈ε2W4.␈↓ α	␈ε"[␈↓ α∪␈ε(B␈↓ α/␈ε"[␈↓ α9␈ε(a␈↓ α]␈ε",␈↓ αm␈ε".␈αε.␈αε.␈↓ β≥␈ε",␈↓ β-␈ε(a␈↓ βO␈ε"]␈↓ βe␈ε"full?]␈α→If␈↓ ∧n␈ε(C␈↓ ¬␈ε"[␈↓ ¬∃␈ε(a␈↓ ¬9␈ε",␈↓ ¬I␈ε".␈αε.␈αε.␈↓ ¬y␈ε",␈↓ ε	␈ε(a␈↓ ε+␈ε"]␈α
<␈α
3␈ε6␈αλ↓␈↓ π→␈ε"4␈↓ πh␈ε",␈αgo␈αto␈αW6.
␈β∂]␈↓ αL␈ε%1␈↓ β@␈ε+r␈↓ ¬(␈ε%1␈↓ ε≤␈ε+r
␈β⊂∧␈↓ ↓@␈ε2W5.␈↓ α	␈ε"[Increase␈↓ β≥␈ε(r␈↓ β.␈ε".]␈α→Set␈↓ ∧→␈ε(r␈↓ ∧3␈ε6␈ ␈↓ ∧a␈ε(r␈↓ ∧z␈ε"+␈αλ1␈αand␈αreturn␈αto␈αW3.
␈β⊂8␈↓ ↓@␈ε2W6.␈↓ α	␈ε"[Set␈↓ αP␈ε(U␈↓ α⎇␈ε".]␈α∃Increase␈αthe␈αv␈α}alue␈αof␈↓ ¬q␈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␈α
v␈α}alue.
␈β⊂D␈↓ αi␈ε+n␈↓ ε+␈ε%1␈↓ π∨␈ε+r
␈β⊂c␈↓ α␈ε"Set␈↓ αJ␈ε(U␈↓ β↓␈ε6␈ ␈↓ β/␈ε(V␈↓ βW␈ε",␈αwhere␈↓ ∧U␈ε(V␈↓ ¬	␈ε"is␈αde|ned␈αin␈αLemma␈αT␈αabo␈α␈v␈α␈e.
␈β⊂o␈↓ αc␈ε+n␈↓ βE␈ε+k␈↓ ∧k␈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↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈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 ␈↓ π←␈ε+r␈↓ πm␈ε9␈␈ε%␈α␈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	↓␈↓ πM␈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␈↓ βi␈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␈↓ ∧α␈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␈↓ λK␈ε(U␈↓ λx␈ε6i␈↓ 	∧␈ε6R␈↓ 	A␈ε/is␈α∞in|nite,␈α∂it␈α∞is
␈β¬∃␈↓ λd␈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␈↓ ∧B␈ε"]␈α∞den␈α↓ote␈α
the␈α∞(possibly␈α
empt␈α␈y)␈α
subsequence␈α∞of␈ε6␈α
h␈↓ 
$␈ε(U␈↓ 
Q␈ε6i␈ε"␈α
con-
␈β¬v␈↓ β?␈ε%1␈↓ ∧3␈ε+r␈↓ 
=␈ε+n
␈βε∃␈↓ ↓H␈ε"taining␈α∞precisely␈α∞th␈α↓ose␈α∞elemen␈α␈ts␈↓ ¬O␈ε(U␈↓ ε
␈ε"that,␈α∂for␈α∞all␈↓ πS␈ε(j␈↓ πr␈ε6∀␈↓ λ#␈ε(r␈↓ λ4␈ε",␈α∞belong␈α∞to␈α∞subsequence
␈βε!␈↓ ¬h␈ε+n
␈βε@␈↓ ↓H␈ε6h␈↓ ↓T␈ε(U␈↓ α↓␈ε6i␈↓ α
␈ε6R␈↓ αD␈ε"if␈↓ αf␈ε(a␈↓ β∩␈ε"=␈α
1␈αand␈αdo␈αn␈α↓ot␈αbelong␈αto␈αsubsequence␈ε6␈αh␈↓ λ␈ε(U␈↓ λ8␈ε6i␈↓ λD␈ε6R␈↓ λ{␈ε"if␈↓ 	≥␈ε(a␈↓ 	I␈ε"=␈α
0.
␈βεL␈↓ ↓l␈ε+n␈↓ α)␈ε+j␈↓ αy␈ε+j␈↓ λ$␈ε+n␈↓ λ`␈ε+j␈↓ 	0␈ε+j
␈βεk␈↓ α␈ε"It␈α∞su}ces␈α∞to␈α∞pro␈α␈v␈α␈e,␈α∂for␈α∞all␈↓ ¬:␈ε(r␈↓ ¬X␈ε6∃␈ε"␈α
1␈α∞and␈α∞all␈α∞pairs␈α∞of␈α∞binary␈α∞n␈α␈um␈α␈bers␈↓ 
0␈ε(a␈↓ 
Z␈ε".␈αε.␈αε.␈↓ 
␈ε(a
␈βεx␈↓ 
C␈ε%1␈↓ ≥␈ε+r
␈βπ⊂␈↓ εb␈ε9␈␈↓ π␈ε+r
␈βπ⊗␈↓ ↓H␈ε"and␈↓ α∀␈ε(b␈↓ α:␈ε".␈αε.␈αε.␈↓ αj␈ε(b␈↓ βλ␈ε",␈α∀that␈↓ βy␈ε"P␈↓ ∧⊃␈ε"r␈↓ ∧ ␈ε"(␈↓ ∧,␈ε(U␈↓ ∧l␈ε62␈↓ ¬_␈ε(I␈↓ ¬y␈ε")␈α∪=␈↓ εP␈ε"2␈↓ π!␈ε"with␈α∩respect␈α∩to␈α∩the␈α⊃subsequence
␈βπ#␈↓ α#␈ε%1␈↓ αy␈ε+r␈↓ ∧D␈ε+n␈↓ ¬'␈ε+b␈↓ ¬B␈ε%...␈↓ ¬]␈ε+b␈↓ π@␈ε↓␈␈↓ 	!␈ε↓↓
␈βπ,␈↓ ¬4␈ε'1␈↓ ¬i␈ε-r
␈βπB␈↓ ↓H␈ε(A␈↓ ↓b␈ε"[␈↓ ↓l␈ε(a␈↓ α⊂␈ε",␈↓ α ␈ε".␈αε.␈αε.␈↓ αP␈ε",␈↓ α`␈ε(a␈↓ βα␈ε"],␈α∀whenev␈α␈er␈α∩the␈α∪latter␈α∩is␈α∩in|nite.␈↓ πN␈ε"See␈α∪Eq.␈α∩(30).␈↓ 	A␈ε"F␈α⎇or␈α∩if␈↓ 
1␈ε(r␈↓ 
V␈ε6∃␈↓ ∞␈ε(k␈↓ "␈ε",
␈βπN␈↓ ↓␈␈ε%1␈↓ αs␈ε+r
␈βπm␈↓ ↓H␈ε"the␈α∩in|nite␈α∩sequence␈ε6␈α∩h␈↓ ∧2␈ε(U␈↓ ∧←␈ε6i␈↓ ∧k␈ε6R␈↓ ¬+␈ε"is␈α∩the␈α∩|nite␈α∩union␈α∩of␈α∩the␈α∩disjoin␈α␈t␈α∩subsequences
␈βπy␈↓ ∧K␈ε+n␈↓ ¬π␈ε+k
␈βλ_␈↓ ↓H␈ε(A␈↓ ↓b␈ε"[␈↓ ↓l␈ε(a␈↓ α⊂␈ε",␈↓ α ␈ε".␈αε.␈αε.␈↓ αP␈ε",␈↓ α`␈ε(a␈↓ βα␈ε"]␈α
for␈↓ βR␈ε(a␈↓ ∧β␈ε"=␈α1␈α
and␈↓ ¬→␈ε(a␈↓ ¬G␈ε"=␈α0␈α
or␈α
1␈α
for␈α
1␈ε6␈α∀␈↓ πi␈ε(j␈↓ λ¬␈ε6∀␈↓ λ5␈ε(r␈↓ λF␈ε",␈↓ λ]␈ε(j␈↓ λz␈ε6≤␈↓ 	*␈ε(k␈↓ 	>␈ε";␈α
and␈α
it␈α
follo␈α␈ws
␈βλ$␈↓ ↓␈␈ε%1␈↓ αs␈ε+r␈↓ βe␈ε+k␈↓ ¬,␈ε+j
␈βλ=␈↓ ∧W␈ε9␈␈↓ ∧u␈ε+r
␈βλC␈↓ ↓H␈ε"that␈↓ α⊗␈ε"P␈↓ α.␈ε"r␈↓ α<␈ε"(␈↓ αH␈ε(U␈↓ α␈␈ε62␈↓ β!␈ε(I␈↓ ∧↓␈ε")␈α
=␈↓ ∧E␈ε"2␈↓ ¬⊂␈ε"with␈αrespect␈αto␈ε6␈αh␈↓ π∀␈ε(U␈↓ πA␈ε6i␈↓ πM␈ε6R␈↓ π{␈ε".␈α→(See␈αexercise␈α33.)␈α→This␈αis
␈βλP␈↓ αa␈ε+n␈↓ β0␈ε+b␈↓ βK␈ε%..␈α␈.␈↓ βe␈ε+b␈↓ π-␈ε+n␈↓ πi␈ε+k
␈βλY␈↓ β<␈ε'1␈↓ βr␈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␈↓ αi␈ε"[␈↓ αt␈ε(a␈↓ β↔␈ε",␈↓ β'␈ε".␈αε.␈αε.␈↓ βW␈ε",␈↓ βg␈ε(a␈↓ ∧
␈ε"]␈αden␈α↓ote␈αthe␈αsubsequence␈αof␈ε6␈αh␈↓ πQ␈ε(U␈↓ π}␈ε6i␈ε"␈αthat␈αconsists␈αof␈αthe␈αv␈α}alues
␈β	&␈↓ βπ␈ε%1␈↓ βz␈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"[␈↓ ∧F␈ε(a␈↓ ∧j␈ε",␈↓ ∧z␈ε".␈αε.␈αε.␈↓ ¬*␈ε",␈↓ ¬:␈ε(a␈↓ ¬\␈ε"]␈α∂is␈α∞increased␈α∂by␈α∞one␈α∂in␈α∞step␈α∂W6␈α∞of␈α∂the␈α∞algo-
␈β	Q␈↓ ∧Y␈ε%1␈↓ ¬M␈ε+r
␈β	j␈↓ 
p␈ε+r␈↓ 
}␈ε9␈␈ε%1
␈β	p␈↓ ↓H␈ε"rithm.␈α⊂By␈αthe␈αalgorithm,␈↓ ∧\␈ε(B␈↓ ∧w␈ε"[␈↓ ¬↓␈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␈απ↓␈↓ 
↑␈ε"4
␈β	|␈↓ ¬∀␈ε%1␈↓ ελ␈ε+r
␈β
≠␈↓ ↓H␈ε"elemen␈α␈ts.␈α∂All␈αbut␈α
a␈α|nite␈α
n␈α␈um␈α␈ber␈αof␈αthe␈α
mem␈α␈bers␈αof␈ε(␈α
A␈↓ λ⊗␈ε"[␈↓ λ!␈ε(a␈↓ λD␈ε",␈↓ λT␈ε".␈αε.␈αε.␈↓ 	∧␈ε",␈↓ 	∀␈ε(a␈↓ 	7␈ε"]␈α
come␈αfrom␈α
the
␈β
(␈↓ λ4␈ε%1␈↓ 	'␈ε+r
␈β
F␈↓ ↓H␈ε"subsequences␈↓ β ␈ε(B␈↓ β;␈ε"[␈↓ βF␈ε(a␈↓ βi␈ε",␈↓ βy␈ε".␈αε.␈αε.␈↓ ∧)␈ε",␈↓ ∧9␈ε(a␈↓ ∧\␈ε",␈↓ ∧l␈ε".␈αε.␈αε.␈↓ ¬≤␈ε",␈↓ ¬,␈ε(a␈↓ ¬L␈ε"],␈αwhere␈↓ εT␈ε(a␈↓ π␈ε"=␈α
0␈αor␈α1␈αfor␈↓ λN␈ε(r␈↓ λh␈ε"<␈↓ 	⊗␈ε(j␈↓ 	1␈ε6∀␈ε(␈αt␈ε".
␈β
S␈↓ βY␈ε%1␈↓ ∧L␈ε+r␈↓ ¬?␈ε+t␈↓ εg␈ε+j
␈β
r␈↓ α␈ε"No␈α␈w␈α⊃assume␈α⊃that␈ε(␈α⊃A␈↓ ∧R␈ε"[␈↓ ∧\␈ε(a␈↓ ¬␈ε",␈↓ ¬⊂␈ε".␈αε.␈αε.␈↓ ¬@␈ε",␈↓ ¬P␈ε(a␈↓ ¬r␈ε"]␈α⊃is␈α⊃in|nite,␈α∩and␈α⊃let␈ε(␈α⊃A␈↓ λ↑␈ε"[␈↓ λh␈ε(a␈↓ 	␈ε",␈↓ 	≤␈ε".␈αε.␈αε.␈↓ 	L␈ε",␈↓ 	\␈ε(a␈↓ 	}␈ε"]␈α∪=␈ε6␈α∩h␈↓ 
]␈ε(U␈↓ ⊗␈ε6i␈ε",
␈β
}␈↓ ∧o␈ε%1␈↓ ¬c␈ε+r␈↓ λ{␈ε%1␈↓ 	o␈ε+r␈↓ 
u␈ε+s
␈βλ␈↓ β␈ε-n
␈β↔␈↓ λT␈ε+r␈↓ 	.␈ε+q␈↓ 
d␈ε+q␈↓ 
t␈ε%+1
␈β≥␈↓ ↓H␈ε"where␈↓ α/␈ε(s␈↓ α[␈ε"<␈↓ β	␈ε(s␈↓ β5␈ε"<␈↓ βc␈ε(s␈↓ ∧∞␈ε6∀␈↓ ∧<␈ε6↓␈αε↓␈αε↓␈↓ ∧l␈ε".␈α⊂If␈↓ ¬*␈ε(N␈↓ ¬X␈ε"is␈αa␈αlarge␈αin␈α␈teger,␈αwith␈↓ λB␈ε"4␈↓ λn␈ε6∀␈↓ 	≤␈ε"4␈↓ 	I␈ε"<␈↓ 	w␈ε(N␈↓ 
$␈ε6∀␈↓ 
R␈ε"4␈↓ "␈ε",
␈β)␈↓ α@␈ε%0␈↓ β~␈ε%1␈↓ βt␈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␈↓ εQ␈ε(k␈↓ εv␈ε"<␈↓ π+␈ε(N␈↓ π↑␈ε"for␈α⊂which␈↓ 	ε␈ε(U␈↓ 	N␈ε"is␈α⊂in␈↓ 
$␈ε(I␈↓ ∀␈ε"is
␈βT␈↓ 	∨␈ε+s␈↓ 
3␈ε+b␈↓ 
N␈ε%...␈↓ 
h␈ε+b
␈β↑␈↓ 	-␈ε-k␈↓ 
?␈ε'1␈↓ 
u␈ε-r
␈βs␈↓ ↓H␈ε"(except␈αfor␈α|nitely␈αman␈α␈y␈αelemen␈α␈ts␈αat␈αthe␈αbeginning␈αof␈αthe␈αsubsequence)
␈β1␈↓ ∧F␈ε(↔␈↓ ∧[␈ε"(␈↓ ∧g␈ε(N␈↓ ¬
␈ε")␈α
=␈↓ ¬N␈ε(↔␈↓ ¬c␈ε"(␈↓ ¬o␈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␈↓ ε@␈ε(B␈↓ ε\␈ε"[␈↓ εf␈ε(a␈↓ π
␈ε",␈↓ π~␈ε".␈αε.␈αε.␈↓ πJ␈ε",␈↓ πZ␈ε(a␈↓ πz␈ε"]␈α∂listed␈α∂abo␈α␈v␈α␈e␈α∞in␈α∂which␈↓ 
u␈ε(U
␈β{␈↓ εy␈ε%1␈↓ πm␈ε+t␈↓ ∞␈ε+s
␈β
¬␈↓ ≤␈ε-k
␈β
~␈↓ ↓H␈ε"appears␈α⊂for␈α⊃some␈↓ βl␈ε(k␈↓ ∧∩␈ε"<␈↓ ∧H␈ε(N␈↓ ∧k␈ε";␈↓ ¬	␈ε(N␈↓ ¬G␈ε"is␈α⊃the␈α⊃n␈α␈um␈α␈ber␈α⊃of␈α⊃v␈α}alues␈α⊂of␈↓ 	π␈ε(k␈↓ 	,␈ε"with␈↓ 
β␈ε(U␈↓ 
L␈ε"in␈α⊂the
␈β
'␈↓ ¬'␈ε+j␈↓ 
≤␈ε+s
␈β
0␈↓ 
*␈ε-k
␈β
E␈↓ ↓H␈ε"corresponding␈α
subsequence;␈αand␈↓ ¬C␈ε(↔␈↓ ¬X␈ε"(␈↓ ¬d␈ε(N␈↓ ε∩␈ε")␈α
is␈αthe␈α
n␈α␈um␈α␈ber␈αof␈α
such␈α
v␈α}alues␈αthat␈α
are␈α
also
␈β
R␈↓ εβ␈ε+j
␈β
q␈↓ ↓H␈ε"in␈↓ ↓r␈ε(I␈↓ αR␈ε".␈α⊂Therefore␈αby␈αLemma␈αT␈↓ ¬←␈ε",
␈β
⎇␈↓ α␈ε+b␈↓ α≤␈ε%..␈α␈.␈↓ α6␈ε+b
␈β∞π␈↓ α
␈ε'1␈↓ αC␈ε-r
␈β∞#␈↓ βZ␈ε9␈␈↓ βx␈ε+r␈↓ ε_␈ε9␈␈↓ ε6␈ε+r␈↓ 	5␈ε9␈␈↓ 	R␈ε+r
␈β∞+␈↓ α:␈ε6j␈↓ αD␈ε(↔␈↓ αY␈ε"(␈↓ αe␈ε(N␈↓ βλ␈ε")␈ε6␈αλ␈␈↓ βH␈ε"2␈↓ ∧π␈ε(N␈↓ ∧*␈ε6j␈↓ ∧>␈ε"=␈ε6␈α
j␈↓ ∧v␈ε(↔␈↓ ¬␈ε"(␈↓ ¬↔␈ε(N␈↓ ¬F␈ε")␈ε6␈αλ␈␈↓ εε␈ε"2␈↓ εE␈ε(N␈↓ ε⎇␈ε"+␈↓ π)␈ε6↓␈αε↓␈αε↓␈↓ π[␈ε"+␈↓ λπ␈ε(↔␈↓ λ≤␈ε"(␈↓ λ(␈ε(N␈↓ λc␈ε")␈ε6␈αλ␈␈↓ 	#␈ε"2␈↓ 	b␈ε(N␈↓ 
≤␈ε6j
␈β∞7␈↓ ¬6␈ε%1␈↓ εd␈ε%1␈↓ λF␈ε+m␈↓ 
␈ε+m
␈β∞d␈↓ ε_␈ε9␈␈↓ ε6␈ε+r␈↓ 	I␈ε9␈␈↓ 	f␈ε+r
␈β∞l␈↓ ∧>␈ε6∀␈α
j␈↓ ∧v␈ε(↔␈↓ ¬␈ε"(␈↓ ¬↔␈ε(N␈↓ ¬F␈ε")␈ε6␈αλ␈␈↓ εε␈ε"2␈↓ εE␈ε(N␈↓ εu␈ε6j␈ε"␈αλ+␈↓ π3␈ε6↓␈αε↓␈αε↓␈↓ πe␈ε"+␈ε6␈αλj␈↓ λ≠␈ε(↔␈↓ λ0␈ε"(␈↓ λ<␈ε(N␈↓ λw␈ε")␈ε6␈αλ␈␈↓ 	7␈ε"2␈↓ 	v␈ε(N␈↓ 
0␈ε6j
␈β∞x␈↓ ¬6␈ε%1␈↓ εd␈ε%1␈↓ λZ␈ε+m␈↓ 
∀␈ε+m
␈β∂%␈↓ λε␈ε+q␈↓ λ⊗␈ε9␈␈↓ λ3␈ε+r␈↓ λA␈ε%+1␈↓ 	9␈ε+q␈↓ 	I␈ε%+1
␈β∂-␈↓ ∧>␈ε6∀␈ε(␈α
m␈ε6␈α
∀␈ε"␈α
1␈αλ+␈αλ2␈αλ+␈αλ4␈αλ+␈↓ π⊗␈ε6↓␈αε↓␈αε↓␈↓ πH␈ε"+␈↓ πt␈ε"2␈↓ λy␈ε"<␈↓ 	'␈ε"2␈↓ 	x␈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␈↓ 
3␈ε",␈↓ 
I␈ε(U␈↓ ∀␈ε"is
␈β∂o␈↓ 
b␈ε+s
␈β∂y␈↓ 
p␈ε-N
␈β⊂∞␈↓ ↓H␈ε"in␈↓ ↓r␈ε(B␈↓ α
␈ε"[␈↓ α_␈ε(a␈↓ α;␈ε",␈↓ αK␈ε".␈αε.␈αε.␈↓ α{␈ε",␈↓ β␈ε(a␈↓ β+␈ε"]␈αfor␈αsome␈ε(␈αt␈ε6␈α∀␈↓ ¬→␈ε(q␈↓ ¬4␈ε"+␈αλ1.
␈β⊂≠␈↓ α+␈ε%1␈↓ β≡␈ε+t
␈β⊂9␈↓ α␈ε"W␈α⎇e␈αha␈α␈v␈α␈e␈αpro␈α␈v␈α␈ed␈αthat
␈β⊂t␈↓ ¬:␈ε9␈␈↓ ¬X␈ε+r␈↓ ε;␈ε+q␈↓ εK␈ε%+1
␈β⊂u␈↓ λ
␈ε6p
␈β⊂x␈↓ λ(␈∧⊂xλ(α#
␈β⊂|␈↓ βe␈ε6j␈↓ βo␈ε(↔␈↓ ∧∧␈ε"(␈↓ ∧⊂␈ε(N␈↓ ∧3␈ε")/␈↓ ∧Q␈ε(N␈↓ ∧|␈ε6␈␈↓ ¬(␈ε"2␈↓ ¬g␈ε6j␈α
∀␈↓ ε)␈ε"2␈↓ εy␈ε"/␈↓ π␈ε(N␈↓ π8␈ε"<␈α
2/␈↓ λ(␈ε(N␈↓ λK␈ε".
␈β⊃¬␈↓ λ⎇␈∧⊃¬λ⎇≠∂
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"159
␈β↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈βα$␈↓ α␈ε"T␈α⎇o␈α
sh␈α↓o␈α␈w␈α∞|nally␈α
that␈α∞sequences␈α
satisfying␈α
De|nition␈α∞R5␈α
exist,␈α∞w␈α␈e␈α
n␈α↓ote␈α
|rst
␈βαO␈↓ ↓H␈ε"that␈α	if␈ε6␈α
h␈↓ α?␈ε(U␈↓ αl␈ε6i␈ε"␈α
is␈α	a␈↓ β?␈ε"[␈↓ βP␈ε"0,␈αε1)␈α	sequence␈α
of␈α
rational␈α	n␈α␈um␈α␈bers␈α
and␈α
if␈ε6␈α
R␈ε"␈α	is␈α
a␈α
computable␈α	sub-
␈βα\␈↓ αX␈ε+n
␈βαz␈↓ ↓H␈ε"sequence␈αλrule␈α	for␈α	a␈ε(␈αλb␈ε"␈α↓-ary␈αλsequence,␈α
w␈α␈e␈αλcan␈α	mak␈α␈e␈ε6␈α	R␈ε"␈αλin␈α␈to␈α	a␈α	computable␈αλsubsequence
␈ββ ␈↓ α)␈ε90␈↓ ∧w␈ε90␈↓ π
␈ε90
␈ββ&␈↓ ↓H␈ε"rule␈↓ α
␈ε6R␈↓ α=␈ε"for␈ε6␈α	h␈↓ α}␈ε(U␈↓ β+␈ε6i␈ε"␈α
by␈α
letting␈↓ ∧a␈ε(f␈↓ ¬␈ε"(␈↓ ¬↔␈ε(x␈↓ ¬<␈ε",␈↓ ¬L␈ε".␈αε.␈αε.␈↓ ¬|␈ε",␈↓ ε␈ε(x␈↓ ε4␈ε")␈α
in␈↓ εq␈ε6R␈↓ π!␈ε"equal␈↓ π|␈ε(f␈↓ λ"␈ε"(␈ε6b␈ε(b␈↓ λL␈ε(x␈↓ λp␈ε6c␈ε",␈↓ 	∞␈ε".␈αε.␈αε.␈↓ 	>␈ε",␈ε6␈αεb␈ε(b␈↓ 	l␈ε(x␈↓ 
∀␈ε6c␈ε")␈α
in␈ε6␈α	R␈ε".␈α∂If
␈ββ2␈↓ β↔␈ε+n␈↓ ¬+␈ε%1␈↓ ε ␈ε+n␈↓ λ∞␈ε+n␈↓ λ`␈ε%1␈↓ 
␈ε+n
␈ββ6␈↓ ∧w␈ε+n
␈ββK␈↓ ∧Y␈ε90
␈ββQ␈↓ ↓H␈ε"the␈↓ αε␈ε"[␈↓ α↔␈ε"0,␈αε1)␈αsequence␈ε6␈α
h␈↓ ∧∧␈ε(U␈↓ ∧1␈ε6i␈↓ ∧=␈ε6R␈↓ ∧o␈ε"is␈α
equidistributed,␈α
so␈αis␈α
the␈ε(␈α
b␈ε"-ary␈α
sequence␈ε6␈αhb␈ε(b␈↓ 
?␈ε(U␈↓ 
l␈ε6ciR␈ε".
␈ββ]␈↓ ∧≥␈ε+n␈↓ 
X␈ε+n
␈ββ|␈↓ ↓H␈ε"No␈α␈w␈αλthe␈α	set␈α	of␈α	all␈αλcomputable␈α	subsequence␈α	rules␈α	for␈ε(␈αλb␈ε"␈α↓-ary␈α	sequences,␈α	for␈α	all␈αλv␈α}alues
␈β∧'␈↓ ↓H␈ε"of␈ε(␈α	b␈ε",␈α
is␈α	coun␈α␈table␈α	(since␈α	only␈α
coun␈α␈tably␈α	man␈α␈y␈α	e{ectiv␈α␈e␈α	algorithms␈α	are␈α
possible),␈α	so
␈β∧R␈↓ ↓H␈ε"they␈αma␈α␈y␈αbe␈αlisted␈αin␈αsome␈αsequence␈↓ ελ␈ε6R␈↓ ε5␈ε",␈↓ εJ␈ε6R␈↓ εw␈ε",␈↓ π␈ε".␈αε.␈αε.␈↓ π<␈ε";␈αtherefore␈αAlgorithm␈αW␈αde|nes
␈β∧←␈↓ ε$␈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␈α
pro␈α↓of␈αof␈αthe␈αexistence␈αof␈α
such␈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;␈α⊂w␈α␈e␈α∞ha␈α␈v␈α␈e␈α∞merely␈α∞stum␈α␈bled␈α∂on␈α∞the␈α∞fact␈α∞that
␈βπ/␈↓ ↓H␈ε"the␈α⊂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⊂algo-
␈βπZ␈↓ ↓H␈ε"rithm.␈α∩In␈αother␈α
w␈α␈ords,␈α
there␈αis␈α
n␈α↓o␈α
e{ectiv␈α␈e␈αalgorithm␈α
to␈α
select␈αthe␈↓ 	k␈ε(j␈↓ 	|␈ε"th␈αcomput-
␈βλ¬␈↓ ↓H␈ε"able␈α
subsequence␈αrule␈↓ ∧!␈ε6R␈↓ ∧L␈ε";␈αthis␈αhappens␈α
because␈αthere␈α
is␈αn␈α↓o␈α
e{ectiv␈α␈e␈αalgorithm␈α
to
␈βλ∩␈↓ ∧=␈ε+j
␈βλ1␈↓ ↓H␈ε"determine␈αif␈α
a␈αcomputational␈α
meth␈α↓od␈αev␈α␈er␈αterminates.␈α≠(W␈α⎇e␈αshall␈αreturn␈α
to␈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␈α
e{ectiv␈α␈e␈αalgorithm,␈α
a␈αsequence␈α
that␈α
satis|es␈αDe|nition␈α
R5␈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␈αW6␈α
of␈α
Algorithm␈αW␈↓ εy␈ε",␈α
so␈α
that␈α
it␈α
sets␈↓ λw␈ε(U␈↓ 	0␈ε6␈ ␈↓ 	←␈ε(V␈↓ 
<␈ε"instead
␈β
⊗␈↓ 	⊂␈ε+n␈↓ 	u␈ε+k␈↓ 
¬␈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␈↓ λ2␈ε(a␈↓ λU␈ε",␈↓ λk␈ε".␈αε.␈αε.␈↓ 	≠␈ε",␈↓ 	1␈ε(a␈↓ 	S␈ε",␈αw␈α␈e␈αcan␈αsh␈α↓o␈α␈w
␈β
B␈↓ απ␈ε+k␈↓ λE␈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␈↓ λZ␈ε(X␈↓ 	∞␈ε".␈αε.␈αε.␈↓ 	>␈ε")␈↓ 	[␈ε/.␈α≥Under␈α⊂this
␈βu␈↓ β
␈ε+n␈↓ λI␈ε%0␈↓ λx␈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␈↓ βλ␈ε6S␈↓ β.␈ε"be␈α
an␈α∞e{ectiv␈α␈e␈α
algorithm␈α
that␈α∞determines␈α
an␈α∞in|nite␈α
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of
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h␈↓ ¬∨␈ε(s␈↓ ¬D␈ε6i␈ε",␈αwhere␈αthe␈α
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␈ε(X␈↓ 
?␈ε6i␈ε/,␈α⊃such
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F␈ε(a␈↓ 
i␈ε",␈↓ 
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Y␈ε%1␈↓ ∩␈ε%2
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p␈ε"(32)
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is␈α
a␈α
measurable␈α	set:␈αEach
␈β∧3␈↓ πβ␈ε%1␈↓ π'␈ε%2␈↓ λ↓␈ε+r
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␈β∧↑␈↓ β6␈ε%1␈↓ βZ␈ε%2␈↓ ∧4␈ε+r␈↓ λ∀␈ε%0␈↓ λC␈ε%1␈↓ 	∃␈ε%2
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∧␈ε(s␈↓ 
&␈ε",␈↓ 
;␈ε(s␈↓ 
]␈ε",␈↓ 
r␈ε".␈αε.␈αε.␈↓ "␈ε",
␈β¬	␈↓ 
∃␈ε%0␈↓ 
L␈ε%1
␈β¬(␈↓ ↓H␈ε(s␈↓ ↓u␈ε",␈αand␈αrule␈ε6␈αR␈ε"␈α
determines␈αa␈αsubsequence␈αof␈↓ π
␈ε(X␈↓ πG␈ε",␈↓ π]␈ε(X␈↓ λ→␈ε",␈↓ λ/␈ε".␈αε.␈αε.␈↓ λ←␈ε",␈↓ λu␈ε(X␈↓ 	G␈ε"such␈αthat␈↓ 
g␈ε(X
␈β¬4␈↓ ↓Y␈ε+m␈↓ π(␈ε+s␈↓ π{␈ε+s␈↓ 	∪␈ε+s␈↓ ¬␈ε+s
␈β¬>␈↓ π6␈ε'0␈↓ λ	␈ε'1␈↓ 	!␈ε-m␈↓ ∪␈ε-m
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∂␈ε(Y␈↓ 
4␈ε(Y␈↓ 
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␈β¬←␈↓ 
#␈ε%0␈↓ 
I␈ε%1␈↓ ≤␈ε%2
␈β¬}␈↓ ↓H␈ε"such␈α∩that␈↓ αu␈ε(Y␈↓ β>␈ε"=␈↓ βw␈ε(X␈↓ ∧F␈ε"for␈α∪0␈ε6␈α∃∀␈↓ ¬e␈ε(k␈↓ ε∂␈ε6∀␈ε(␈α∃m␈ε"␈α∩also␈α∪belongs␈α∪to␈↓ 	α␈ε(T␈↓ 	≤␈ε"(␈↓ 	(␈ε(a␈↓ 	L␈ε(a␈↓ 	v␈ε".␈αε.␈αε.␈↓ 
&␈ε(a␈↓ 
H␈ε"),␈α∀and
␈βε␈↓ β
␈ε+s␈↓ ∧∃␈ε+s␈↓ 	;␈ε%1␈↓ 	←␈ε%2␈↓ 
9␈ε+r
␈βε∀␈↓ β_␈ε-k␈↓ ∧#␈ε-k
␈βε)␈↓ ↓H␈ε"this␈α⊃is␈α∩a␈α∩measurable␈α∩set␈α⊃consisting␈α∩of␈α∩the␈α∩|nite␈α⊃union␈α∩of␈α∩dy␈α␈adic␈α⊃subin␈α␈terv␈α}als
␈βεU␈↓ ↓H␈ε(I␈↓ α&␈ε".␈α&Since␈α∪there␈α∪are␈α∪only␈α∪coun␈α␈tably␈α∪man␈α␈y␈α∪such␈α∪dy␈α␈adic␈α∪in␈α␈terv␈α}als,␈α∃w␈α␈e␈α∪see
␈βεa␈↓ ↓V␈ε+b␈↓ ↓r␈ε%.␈α␈..␈↓ α␈ε+b
␈βεk␈↓ ↓c␈ε'1␈↓ α→␈ε-t
␈βπ␈↓ ↓H␈ε"that␈↓ α_␈ε(T␈↓ α2␈ε"(␈↓ α>␈ε(a␈↓ αb␈ε(a␈↓ β␈ε".␈αε.␈αε.␈↓ β<␈ε(a␈↓ β↑␈ε")␈α∂is␈α∞a␈α∂coun␈α␈table␈α∞union␈α∂of␈α∂dy␈α␈adic␈α∞in␈α␈terv␈α}als,␈α⊂and␈α∞it␈α∂is␈α∞therefore
␈βπ␈↓ αQ␈ε%1␈↓ αu␈ε%2␈↓ βO␈ε+r
␈βπ+␈↓ ↓H␈ε"measurable.␈α!F␈α⎇urtherm␈α↓ore,␈α∪this␈α∩argumen␈α␈t␈α⊃can␈α∩be␈α∩extended␈α⊃to␈α∩sh␈α↓o␈α␈w␈α∩that␈α⊃the
␈βπV␈↓ ↓H␈ε"measure␈αλof␈↓ αw␈ε(T␈↓ β⊂␈ε"(␈↓ β≤␈ε(a␈↓ βF␈ε".␈αε.␈αε.␈↓ βv␈ε(a␈↓ ∧J␈ε"0)␈α	equals␈αλthe␈α	measure␈αλof␈↓ πC␈ε(T␈↓ π\␈ε"(␈↓ πh␈ε(a␈↓ λ∩␈ε".␈αε.␈αε.␈↓ λB␈ε(a␈↓ 	⊗␈ε"1),␈α	since␈α	the␈αλlatter
␈βπc␈↓ β/␈ε%1␈↓ ∧	␈ε+r␈↓ ∧↔␈ε9␈␈ε%1␈↓ π{␈ε%1␈↓ λU␈ε+r␈↓ λc␈ε9␈␈ε%1
␈βλ↓␈↓ ↓H␈ε"is␈αa␈αunion␈αof␈αdy␈α␈adic␈αin␈α␈terv␈α}als␈αobtained␈αfrom␈αthe␈αformer␈αby␈αrequiring␈αthat␈↓ 
K␈ε(Y␈↓ λ␈ε"=
␈βλ∞␈↓ 
`␈ε+s
␈βλ↔␈↓ 
n␈ε-k
␈βλ-␈↓ ↓H␈ε(X␈↓ α⊂␈ε"for␈α
0␈ε6␈α
∀␈↓ β∪␈ε(k␈↓ β1␈ε"<␈ε(␈α
m␈ε"␈αand␈↓ ∧P␈ε(Y␈↓ ¬⊗␈ε6≤␈↓ ¬D␈ε(X␈↓ ε
␈ε".␈α⊂No␈α␈w␈αsince
␈βλ9␈↓ ↓f␈ε+s␈↓ ∧e␈ε+s␈↓ ¬b␈ε+s
␈βλC␈↓ ↓t␈ε-k␈↓ ∧s␈ε-m␈↓ ¬p␈ε-m
␈βλn␈↓ β%␈ε(T␈↓ β?␈ε"(␈↓ βK␈ε(a␈↓ βu␈ε".␈αε.␈αε.␈↓ ∧%␈ε(a␈↓ ∧y␈ε"0)␈ε6␈αλ[␈↓ ¬A␈ε(T␈↓ ¬Z␈ε"(␈↓ ¬f␈ε(a␈↓ ε⊂␈ε".␈αε.␈αε.␈↓ ε@␈ε(a␈↓ π∀␈ε"1)␈ε6␈α
∩␈↓ πj␈ε(T␈↓ λ∧␈ε"(␈↓ λ⊂␈ε(a␈↓ λ:␈ε".␈αε.␈αε.␈↓ λj␈ε(a␈↓ 	9␈ε"),
␈βλz␈↓ β↑␈ε%1␈↓ ∧8␈ε+r␈↓ ∧E␈ε9␈␈ε%1␈↓ ¬y␈ε%1␈↓ εS␈ε+r␈↓ εa␈ε9␈␈ε%1␈↓ λ#␈ε%1␈↓ λ⎇␈ε+r␈↓ 	␈ε9␈␈ε%␈α␈1
␈β	/␈↓ ↓H␈ε"the␈αmeasure␈α
of␈↓ β>␈ε(T␈↓ βW␈ε"(␈↓ βc␈ε(a␈↓ ∧π␈ε(a␈↓ ∧1␈ε".␈αε.␈αε.␈↓ ∧a␈ε(a␈↓ ¬β␈ε")␈α
is␈α
at␈αm␈α↓ost␈α
one-half␈α
the␈αmeasure␈α
of␈↓ 	H␈ε(T␈↓ 	a␈ε"(␈↓ 	m␈ε(a␈↓ 
↔␈ε".␈αε.␈αε.␈↓ 
G␈ε(a␈↓ ⊗␈ε").
␈β	;␈↓ βv␈ε%1␈↓ ∧~␈ε%2␈↓ ∧t␈ε+r␈↓ 
␈ε%1␈↓ 
Z␈ε+r␈↓ 
h␈ε9␈␈ε%1
␈β	Z␈↓ ↓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
1␈↓ ↓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
\␈↓ ↓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
␈βπ␈↓ ↓H␈ε"pro␈α↓of␈α
of␈α
this␈αfact,␈α
and␈αthey␈α
serv␈α␈e␈α
to␈αpro␈α␈vide␈α
probabilit␈α␈y␈α
estimates␈αthat␈α
are␈α
useful
␈β2␈↓ ↓H␈ε"in␈αman␈α␈y␈αother␈αproblems.
␈βB␈↓ ¬j␈ε↓S
␈β]␈↓ α␈ε"F␈α⎇or␈α
0␈α
<␈ε(␈α∂␈ε"␈α<␈α
1,␈α∞let␈↓ ∧H␈ε(B␈↓ ∧c␈ε"(␈↓ ∧o␈ε(r␈↓ ¬␈ε",␈ε(␈αε∂␈ε")␈α
be␈↓ ε∞␈ε(T␈↓ ε'␈ε"(␈↓ ε3␈ε(a␈↓ ε]␈ε".␈αε.␈αε.␈↓ π
␈ε(a␈↓ π0␈ε"),␈α
where␈α∞the␈α
union␈α∞is␈α
tak␈α␈en␈α
o␈α␈v␈α␈er
␈βj␈↓ εF␈ε%1␈↓ π ␈ε+r
␈β	␈↓ ↓H␈ε"all␈α
binary␈αn␈α␈um␈α␈bers␈↓ β⎇␈ε(a␈↓ ∧'␈ε".␈αε.␈αε.␈↓ ∧W␈ε(a␈↓ ¬ε␈ε"for␈α
which␈α
the␈α
n␈α␈um␈α␈ber␈↓ πj␈ε(↔␈↓ π␈␈ε"(␈↓ λ␈ε(r␈↓ λ≤␈ε")␈α
of␈αzeros␈α
am␈α↓ong␈↓ 
0␈ε(a␈↓ 
Z␈ε".␈αε.␈αε.␈↓ 
␈ε(a
␈β∃␈↓ ∧⊂␈ε%1␈↓ ∧j␈ε+r␈↓ 
C␈ε%1␈↓ ≥␈ε+r
␈β4␈↓ ↓H␈ε"satis|es
␈βZ␈↓ ε∂␈ε%1
␈β←␈↓ ¬⊂␈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".
␈βp␈↓ ε∂␈ε%2
␈βs␈↓ ε∂␈∧sε∂α⊂
␈β{␈↓ π|␈ε↓␈␈↓ λ≠␈ε↓↓
␈β␈␈↓ πP␈ε↓P
␈β
⊃␈↓ λ␈ε+r
␈β
~␈↓ ↓H␈ε"The␈αn␈α␈um␈α␈ber␈αof␈αsuch␈αbinary␈αn␈α␈um␈α␈bers␈αis␈↓ ε4␈ε(C␈↓ εQ␈ε"(␈↓ ε]␈ε(r␈↓ εm␈ε",␈ε(␈αε∂␈ε")␈α
=␈↓ λ5␈ε"summed␈αo␈α␈v␈α␈er␈αthe␈αv␈α}alues
␈β
+␈↓ λ
␈ε+k
␈β
A␈↓ βC␈ε%1
␈β
E␈↓ ↓H␈ε"of␈↓ ↓t␈ε(k␈↓ α⊗␈ε"with␈ε6␈α∞j␈↓ αt␈ε(k␈↓ β∩␈ε6␈␈↓ βV␈ε(r␈↓ βg␈ε6j␈α
∃␈ε"␈α∞1␈α	+␈ε(␈α
∂␈↓ ¬λ␈ε(r␈↓ ¬_␈ε".␈α⊗A␈α∂suitable␈α∞estimate␈α∞for␈α∞the␈α∞tail␈α∞of␈α∞the␈α∞bin␈α↓omial
␈β
V␈↓ βC␈ε%2
␈β
Y␈↓ βC␈∧
YβCα⊂
␈β
p␈↓ ↓H␈ε"distribution␈α
can␈αbe␈α
obtained␈αby␈αthe␈α
follo␈α␈wing␈↓ π∩␈ε"standard␈α
trick:␈αLet␈ε(␈α
x␈ε"␈αand␈ε(␈α
p␈ε"␈αbe␈α
an␈α␈y
␈β∞≤␈↓ ↓H␈ε"positiv␈α␈e␈αn␈α␈um␈α␈bers␈αless␈αthan␈α1,␈αlet␈↓ ¬N␈ε(q␈↓ ¬k␈ε"=␈α
1␈ε6␈αλ␈␈ε(␈αλp␈ε",␈αand␈αlet␈ε(␈αs␈ε"␈α
=␈α
(␈ε(p␈ε"␈αλ+␈ε(␈αλ∂␈ε")␈↓ 	7␈ε(r␈↓ 	G␈ε".␈α⊂Then
␈β∞I␈↓ 
_␈ε↓⊂␈↓ ↓␈ε↓⊃
␈β∞P␈↓ ↓M␈ε↓X␈↓ βx␈ε↓X␈↓ εv␈ε↓X
␈β∞Y␈↓ α␈ε↓∩␈↓ α6␈ε↓∪␈↓ ∧7␈ε↓∩␈↓ ∧a␈ε↓∪␈↓ π6␈ε↓∩␈↓ π`␈ε↓∪␈↓ ∪␈ε+r
␈β∞Z␈↓ α#␈ε(r␈↓ ∧O␈ε(r␈↓ πM␈ε(r␈↓ 
p␈ε(p
␈β∞k␈↓ α↑␈ε+k␈↓ ββ␈ε+r␈↓ β⊃␈ε9␈␈↓ β.␈ε+k␈↓ ¬	␈ε+k␈↓ ¬.␈ε+r␈↓ ¬<␈ε9␈␈↓ ¬Z␈ε+k␈↓ ε␈ε+s␈ε9␈␈↓ ε,␈ε+k␈↓ λλ␈ε+k␈↓ λ-␈ε+r␈↓ λ;␈ε9␈␈↓ λX␈ε+k␈↓ λ␈␈ε+s␈ε9␈␈↓ 	*␈ε+k␈↓ 
λ␈ε+s
␈β∞r␈↓ αL␈ε(p␈↓ αp␈ε(q␈↓ βJ␈ε6∀␈↓ ∧w␈ε(p␈↓ ¬≠␈ε(q␈↓ ¬l␈ε(x␈↓ εH␈ε6∀␈↓ πv␈ε(p␈↓ λ~␈ε(q␈↓ λk␈ε(x␈↓ 	F␈ε"=␈↓ 	t␈ε(x␈↓ 
$␈ε(q␈↓ 
?␈ε"+␈↓ "␈ε".
␈β∂π␈↓ 
o␈∧∂π
oα∀
␈β∂␈↓ α!␈ε(k␈↓ ∧M␈ε(k␈↓ πL␈ε(k␈↓ 
o␈ε(x
␈β∂#␈↓ ↓H␈ε+k␈↓ ↓X␈ε9∃␈ε+s␈↓ βs␈ε+k␈↓ ∧∧␈ε9∃␈ε+␈α␈s␈↓ εq␈ε+k␈↓ π↓␈ε9∃␈ε%0
␈β∂S␈↓ πQ␈ε+s␈↓ λz␈ε+r
␈β∂Y␈↓ ↓H␈ε"By␈α∞elemen␈α␈tary␈α∞calculus,␈α∞the␈α∞minim␈α␈um␈α∞v␈α}alue␈α∞of␈↓ π=␈ε(x␈↓ π`␈ε"(␈↓ πl␈ε(q␈↓ λ	␈ε"+␈ε(␈α	p␈ε"/␈ε(x␈↓ λn␈ε")␈↓ 	↔␈ε"occurs␈α∞when␈ε(␈α∞x␈ε"␈α
=
␈β∂e␈↓ ↓H␈ε↓␈␈↓ αf␈ε↓↓␈↓ αt␈ε↓∞␈↓ β	␈ε↓␈␈↓ ∧*␈ε↓↓
␈β⊂∧␈↓ ↓V␈ε(p␈ε"/(␈ε(p␈ε"␈αλ+␈ε(␈αλ∂␈ε"␈α␈)␈↓ β↔␈ε(q␈↓ β*␈ε"/(␈↓ βH␈ε(q␈↓ βc␈ε6␈␈ε(␈αλ∂␈ε")␈↓ ∧8␈ε",␈αand␈αthis␈αv␈α}alue␈αof␈ε(␈αx␈ε"␈αyields
␈β⊂=␈↓ 	H␈ε+r
␈β⊂H␈↓ ε⊃␈ε↓␈ ␈↓ π1␈ε+p␈ε%+␈ε+∂␈↓ λv␈ε+q␈↓ 	ε␈ε9␈␈ε+␈α␈∂␈↓ 	1␈ε↓!
␈β⊂J␈↓ β>␈ε↓X
␈β⊂S␈↓ ∧$␈ε↓∩␈↓ ∧N␈ε↓∪␈↓ ε(␈ε↓∩␈↓ π≠␈ε↓∪␈↓ πl␈ε↓∩␈↓ λ`␈ε↓∪
␈β⊂T␈↓ ∧<␈ε(r␈↓ εd␈ε(p␈↓ λ'␈ε(q
␈β⊂d␈↓ ∧v␈ε+k␈↓ ¬≤␈ε+r␈↓ ¬)␈ε9␈␈↓ ¬G␈ε+k
␈β⊂l␈↓ ∧d␈ε(p␈↓ ¬λ␈ε(q␈↓ ¬c␈ε6∀␈↓ 	X␈ε".
␈β⊃␈↓ εB␈∧⊃εBαU␈↓ λε␈∧⊃λεαV
␈β⊃ε␈↓ ∧:␈ε(k␈↓ εB␈ε(p␈ε"␈αλ+␈ε(␈αλ∂␈↓ λε␈ε(q␈↓ λ!␈ε6␈␈ε(␈αλ∂
␈β⊃≤␈↓ β∪␈ε+k␈↓ β#␈ε9∃␈ε%(␈ε+p␈ε%+␈ε+∂␈ε%)␈↓ ∧∂␈ε+r
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"161
␈β↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈βα"␈↓ ↓H␈ε"No␈α␈w␈αwhen␈ε(␈α∂␈ε6␈α	∀␈↓ β?␈ε"min␈↓ β{␈ε"(␈ε(p␈ε",␈↓ ∧)␈ε(q␈↓ ∧=␈ε")␈αw␈α␈e␈αha␈α␈v␈α␈e
␈βαi␈↓ αk␈ε↓␈␈↓ ∧␈ε↓↓
␈βαj␈↓ ε
␈ε%2␈↓ εs␈ε%3␈↓ πn␈ε%4
␈βαp␈↓ ¬{␈ε(∂␈↓ εe␈ε(∂␈↓ π`␈ε(∂
␈βαy␈↓ ∧~␈ε+p␈↓ ∧*␈ε%+␈ε+∂
␈ββλ␈↓ αM␈ε"ln␈↓ αz␈ε(p␈↓ β␈ε"/(␈↓ β+␈ε(p␈↓ βE␈ε"+␈ε(␈αλ∂␈ε")␈↓ ∧`␈ε"=␈ε6␈α
␈␈ε(∂␈ε6␈απ␈␈↓ ε)␈ε"+␈↓ π≠␈ε6␈␈↓ λ∨␈ε"+␈↓ λK␈ε6↓␈αε↓␈αε↓␈↓ λ␈␈ε"<␈ε6␈α
␈␈ε(∂␈ε",
␈ββ≤␈↓ ¬x␈∧β≤¬xα&␈↓ εY␈∧β≤εYα6␈↓ πK␈∧β≤πKαH
␈ββ≡␈↓ ε␈␈ε%2␈↓ λα␈ε%3
␈ββ"␈↓ ¬x␈ε"2␈↓ ε␈ε(p␈↓ εY␈ε"6␈↓ εl␈ε(p␈↓ πK␈ε"12␈↓ πp␈ε(p
␈ββQ␈↓ αk␈ε↓␈␈↓ ∧␈ε↓↓
␈ββR␈↓ ε
␈ε%2␈↓ εs␈ε%3␈↓ πn␈ε%4␈↓ 
ε␈ε%2
␈ββX␈↓ ¬{␈ε(∂␈↓ εe␈ε(∂␈↓ π`␈ε(∂␈↓ 	w␈ε(∂
␈ββa␈↓ ∧~␈ε+q␈↓ ∧*␈ε9␈␈ε+∂
␈ββp␈↓ αM␈ε"ln␈↓ αy␈ε(q␈↓ β␈ε"/(␈↓ β*␈ε(q␈↓ βE␈ε6␈␈ε(␈αλ∂␈ε")␈↓ ∧`␈ε"=␈α
+␈ε(∂␈ε6␈απ␈␈↓ ε)␈ε6␈␈↓ π≠␈ε6␈␈↓ λ∨␈ε6␈␈↓ λK␈ε6↓␈αε↓␈αε↓␈↓ λ␈␈ε"<␈ε(␈α
∂␈ε6␈αλ␈␈↓ 
≥␈ε",
␈β∧∧␈↓ ¬x␈∧∧∧¬xα&␈↓ εY␈∧∧∧εYα6␈↓ πK␈∧∧∧πKαH␈↓ 	t␈∧∧∧	tα&
␈β∧ε␈↓ ε␈␈ε%2␈↓ λα␈ε%3
␈β∧	␈↓ ¬x␈ε"2␈↓ ε
␈ε(q␈↓ εY␈ε"6␈↓ εk␈ε(q␈↓ πK␈ε"12␈↓ πo␈ε(q␈↓ 	t␈ε"2␈↓ 
ε␈ε(q
␈β∧S␈↓ ↓H␈ε"so␈αw␈α␈e␈αha␈α␈v␈α␈e␈αthe␈αfollo␈α␈wing␈αbound␈αfor␈αall␈↓ ε+␈ε(r␈↓ εE␈ε">␈α
0␈αand␈α0␈ε6␈α
∀␈ε(␈α
∂␈ε6␈α
∀␈↓ λh␈ε"min␈↓ 	$␈ε"(␈ε(p␈ε",␈↓ 	R␈ε(q␈↓ 	e␈ε"):
␈β¬∃␈↓ ∧T␈ε↓X
␈β¬≡␈↓ ¬:␈ε↓∩␈↓ ¬d␈ε↓∪
␈β¬ ␈↓ ¬R␈ε(r
␈β¬+␈↓ π`␈ε'2
␈β¬0␈↓ ε␈ε+k␈↓ ε2␈ε+r␈↓ ε?␈ε9␈␈↓ ε]␈ε+k␈↓ π6␈ε9␈␈↓ πT␈ε+∂␈↓ πo␈ε+r␈↓ π⎇␈ε%/(2␈↓ λ&␈ε+q␈↓ λ5␈ε%)
␈β¬8␈↓ ¬z␈ε(p␈↓ ε≡␈ε(q␈↓ εy␈ε"<␈↓ π'␈ε(e␈↓ λA␈ε".␈↓ 
p␈ε"(33)
␈β¬Q␈↓ ¬P␈ε(k
␈β¬h␈↓ ∧)␈ε+k␈↓ ∧9␈ε9∃␈ε%(␈ε+p␈ε%+␈ε+∂␈ε%)␈↓ ¬%␈ε+r
␈βε+␈↓ β2␈ε+r␈↓ β?␈ε%+1
␈βε,␈↓ ∂␈ε%1
␈βε1␈↓ ↓H␈ε"But␈↓ α
␈ε(C␈↓ α)␈ε"(␈↓ α5␈ε(r␈↓ αF␈ε",␈ε(␈αε∂␈ε")␈αis␈↓ β ␈ε"2␈↓ βy␈ε"times␈αthis␈αle$-hand␈αquan␈α␈tit␈α␈y,␈αin␈αthe␈αspecial␈αcase␈ε(␈αp␈ε"␈α
=␈↓ 
@␈ε(q␈↓ 
]␈ε"=␈↓ "␈ε",
␈βεB␈↓ ∂␈ε%2
␈βεE␈↓ ∂␈∧εE∂α⊂
␈βε\␈↓ ↓H␈ε"hence␈αby␈α(32)
␈βπ⊂␈↓ λr␈ε'2
␈βπ∃␈↓ ε←␈ε9␈␈↓ ε|␈ε+r␈↓ λH␈ε9␈␈↓ λf␈ε+∂␈↓ 	↓␈ε+r
␈βπ≥␈↓ βZ␈ε(B␈↓ βu␈ε"(␈↓ ∧↓␈ε(r␈↓ ∧∩␈ε",␈ε(␈αε∂␈ε")␈↓ ∧I␈ε"has␈αmeasure␈↓ ε∨␈ε6∀␈↓ εM␈ε"2␈↓ π␈ε(C␈↓ π(␈ε"(␈↓ π4␈ε(r␈↓ πD␈ε",␈ε(␈αε∂␈ε")␈α
<␈α
2␈↓ λ9␈ε(e␈↓ 	⊂␈ε".
␈βπl␈↓ α␈ε"The␈αnext␈αstep␈αis␈αto␈αde|ne
␈βλ;␈↓ 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".
␈βλt␈↓ ¬o␈ε↓P
␈β	∧␈↓ π$␈ε'2
␈β		␈↓ εz␈ε9␈␈↓ π_␈ε+∂␈↓ π2␈ε+k
␈β	∂␈↓ ↓H␈ε"The␈αmeasure␈αof␈↓ βG␈ε(B␈↓ βb␈ε"*(␈↓ ∧␈ε(r␈↓ ∧⊃␈ε",␈ε(␈αε∂␈ε"␈α␈)␈αis␈αat␈αm␈α↓ost␈↓ εY␈ε"2␈↓ εk␈ε(e␈↓ πE␈ε",␈αand␈αthis␈αis␈αthe␈αremainder␈αof␈αa
␈β	!␈↓ ε∃␈ε+k␈↓ ε&␈ε9∃␈↓ εC␈ε+r
␈β	:␈↓ ↓H␈ε"con␈α␈v␈α␈ergen␈α␈t␈αseries,␈αso
␈β	k␈↓ ¬∩␈ε↓␈␈↓ πK␈ε↓↓
␈β
	␈↓ ∧T␈ε"lim␈↓ ¬ ␈ε"measure␈αof␈↓ εV␈ε(B␈↓ εq␈ε"*(␈↓ π∂␈ε(r␈↓ π ␈ε",␈ε(␈αε∂␈ε")␈↓ πc␈ε"=␈α
0.␈↓ 
p␈ε"(34)
␈β
)␈↓ ∧G␈ε+r␈↓ ∧U␈ε9!1
␈β
h␈↓ ↓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.␈↓ λ9␈ε(X␈↓ λg␈ε(X␈↓ 	≤␈ε".␈αε.␈αε.␈↓ 	L␈ε")␈↓ 	y␈ε"leads␈α⊂to␈α∂an
␈β
u␈↓ λW␈ε%0␈↓ 	¬␈ε%1␈↓ 	X␈ε%2
␈β∀␈↓ ↓H␈ε"in|nite␈α	sequence␈ε6␈α	h␈↓ β\␈ε(X␈↓ ∧≠␈ε6iR␈ε"␈αλthat␈α	is␈α	n␈α↓ot␈α	1-distributed,␈α
and␈α	if␈↓ λ5␈ε(↔␈↓ λJ␈ε"(␈↓ λV␈ε(r␈↓ λg␈ε")␈α	den␈α↓otes␈αλthe␈α	n␈α␈um␈α␈ber
␈β ␈↓ βz␈ε+s
␈β*␈↓ ∧λ␈ε-n
␈β?␈↓ ↓H␈ε"of␈αzeros␈αin␈αthe␈α|rst␈↓ β}␈ε(r␈↓ ∧~␈ε"elemen␈α␈ts␈αof␈αthe␈αlatter␈αsequence,␈αthen
␈β	␈↓ εK␈ε%1
␈β∞␈↓ ¬)␈ε6j␈↓ ¬3␈ε(↔␈↓ ¬H␈ε"(␈↓ ¬T␈ε(r␈↓ ¬e␈ε")/␈↓ εβ␈ε(r␈↓ ε≠␈ε6␈␈↓ ε↑␈ε6j␈α
∃␈ε"␈α
2␈ε(∂␈ε",
␈β∨␈↓ εK␈ε%2
␈β"␈↓ εK␈∧"εKα⊂
␈β]␈↓ ↓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
λ␈↓ ↓H␈ε"|nally␈αw␈α␈e␈α|nd␈αthat
␈β
⊃␈↓ ε
␈ε↓[␈↓ εQ␈ε↓\
␈β
3␈↓ ∧J␈ε(N␈↓ ∧m␈ε"(␈ε6R␈ε",␈↓ ¬%␈ε6S␈↓ ¬=␈ε")␈α
=␈↓ π	␈ε(B␈↓ π%␈ε"*(␈↓ πC␈ε(r␈↓ πS␈ε",␈αε1/␈ε(t␈ε"␈α↓);
␈β
d␈↓ ε↓␈ε+t␈ε9␈α↓∃␈ε%␈α␈2␈↓ εG␈ε+r␈↓ εU␈ε9∃␈ε%␈α␈1
␈β
␈␈↓ β+␈ε↓T
␈β∞~␈↓ ↓H␈ε"and,␈α⊃by␈α⊂(34),␈↓ ∧␈ε(B␈↓ ∧&␈ε"*(␈↓ ∧D␈ε(r␈↓ ∧U␈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",␈↓ 
D␈ε6S␈↓ 
\␈ε")␈α⊂has
␈β∞,␈↓ βI␈ε+r␈↓ βV␈ε9∃␈ε%1
␈β∞E␈↓ ↓H␈ε"measure␈αzero.
␈β∞N␈↓ βE␈∧∞NβE≠∂
␈β∂β␈↓ α␈ε"F␈α⎇rom␈α	the␈αλexistence␈α	of␈ε/␈α	binary␈ε"␈α	sequences␈αλsatisfying␈α	De|nition␈α	R6,␈α	w␈α␈e␈α	can␈αλsh␈α↓o␈α␈w
␈β∂.␈↓ ↓H␈ε"the␈α
existence␈α
of␈↓ βM␈ε"[␈↓ β]␈ε"0,␈αε1)␈α∞sequences␈α
that␈α
are␈α
random␈α
in␈α
this␈α
sense.␈α∀F␈α⎇or␈α
details,␈α
see
␈β∂Y␈↓ ↓H␈ε"exercise␈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↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈βα$␈↓ ↓H␈ε"sequence␈α⊃is␈α∩\m␈α↓ore␈α∩random"␈α∩than␈α∩000000000.␈α!Alth␈α↓ough␈α∩it␈α∩is␈α∩true␈α∩that␈α⊃truly
␈βαO␈↓ ↓H␈ε"random␈α⊃sequences␈α⊃will␈α⊃exhibit␈↓ ¬A␈ε"locally␈α⊃n␈α↓onrandom␈α⊃beha␈α␈vior,␈α∪w␈α␈e␈α⊃w␈α␈ould␈α⊃expect
␈βαz␈↓ ↓H␈ε"such␈αbeha␈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␈↓ ¬t␈ε",␈↓ ε
␈ε".␈αε.␈αε.␈↓ ε:␈ε",␈↓ εP␈ε(X␈↓ π␈ε",␈αw␈α␈e␈αcan␈αsa␈α␈y␈↓ λV␈ε"that
␈β∧4␈↓ ¬≡␈ε%1␈↓ ¬c␈ε%2␈↓ εn␈ε+N
␈β∧d␈↓ βw␈ε↓␈␈↓ ∧L␈ε↓↓
␈β∧|␈↓ λY␈ε6p
␈β∧␈␈↓ λw␈∧∧␈λwα#
␈β¬β␈↓ βP␈ε"P␈↓ βh␈ε"r␈↓ ∧¬␈ε(S␈↓ ∧≡␈ε"(␈ε(n␈ε")␈↓ ∧d␈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/␈↓ λw␈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\␈↓ βC␈ε"P␈↓ β[␈ε"r␈↓ βi␈ε"(␈↓ βu␈ε(X␈↓ ∧(␈ε(X␈↓ ¬
␈ε".␈αε.␈αε.␈↓ ¬=␈ε(X␈↓ εT␈ε"=␈↓ πα␈ε(x␈↓ π'␈ε(x␈↓ πQ␈ε".␈αε.␈αε.␈↓ λ↓␈ε(x␈↓ λ(␈ε")␈ε6␈α
→␈ε"␈α
1/␈↓ 	⊂␈ε(b
␈βεh␈↓ ∧∪␈ε+n␈↓ ∧F␈ε+n␈ε%␈α␈+1␈↓ ¬[␈ε+n␈ε%␈α␈+␈↓ ε␈ε+k␈↓ ε≠␈ε9␈␈ε%1␈↓ π⊗␈ε%1␈↓ π;␈ε%2␈↓ λ∃␈ε+k
␈βπ∀␈↓ ¬u␈ε↓␈
␈βπ2␈↓ ↓H␈ε"for␈α∞all␈ε(␈α∂b␈ε"-ary␈α∂n␈α␈um␈α␈bers␈↓ ∧'␈ε(x␈↓ ∧L␈ε(x␈↓ ∧v␈ε".␈αε.␈αε.␈↓ ¬&␈ε(x␈↓ ¬L␈ε".␈↓ εβ␈ε"Cf.␈α∞De|nition␈α∂D␈↓ λ∧␈ε";␈α⊂unfortunately␈α∂a␈α∞sequence
␈βπ?␈↓ ∧;␈ε%1␈↓ ∧`␈ε%2␈↓ ¬:␈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(␈α
b␈ε/␈α↓-ary␈α
sequence␈α
of␈α
length␈↓ εt␈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␈↓ λJ␈ε(X␈↓ 		␈ε6iR␈ε",␈αas␈αin␈αthe␈αpro␈α↓of
␈β∞	␈↓ λh␈ε+s
␈β∞∪␈↓ λv␈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␈↓ ¬e␈ε(X␈↓ ε∪␈ε/,␈↓ ε'␈ε(X␈↓ εV␈ε/,␈↓ εj␈ε".␈αε.␈αε.␈↓ π~␈ε/,␈↓ π.␈ε(X␈↓ πs␈ε/is␈ε"␈α
(␈ε(n␈ε",␈ε(␈αε∂␈ε"␈α␈)␈ε/-random␈α
with␈α	respect
␈β∞v␈↓ εβ␈ε%1␈↓ εE␈ε%2␈↓ πL␈ε+N
␈β∂∀␈↓ ↓H␈ε/to␈α
a␈αset␈αof␈αalgorithms␈ε2␈αA␈↓ ∧:␈ε/,␈αif␈αfor␈α
ev␈α␈ery␈αsubsequence␈↓ πM␈ε(X␈↓ λε␈ε/,␈↓ λ≠␈ε(X␈↓ λU␈ε/,␈↓ λj␈ε".␈αε.␈αε.␈↓ 	~␈ε/,␈↓ 	/␈ε(X␈↓ 	|␈ε/determined
␈β∂!␈↓ πk␈ε+t␈↓ λ9␈ε+t␈↓ 	M␈ε+t
␈β∂*␈↓ πv␈ε'1␈↓ λE␈ε'2␈↓ 	X␈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␈↓ β4␈ε↓␈␈↓ ε?␈ε↓␈
␈β⊂␈↓ β4␈ε↓␈␈↓ ε?␈ε↓␈
␈β⊂∩␈↓ βK␈ε"1␈↓ ε)␈ε"1
␈β⊂!␈↓ β4␈ε↓␈␈↓ ε?␈ε↓␈
␈β⊂*␈↓ βh␈ε(↔␈↓ ∧
␈ε"(␈↓ ∧→␈ε(X␈↓ ∧R␈ε",␈↓ ∧b␈ε".␈αε.␈αε.␈↓ ¬∩␈ε",␈↓ ¬"␈ε(X␈↓ ¬e␈ε")␈ε6␈αλ␈␈↓ εU␈ε6∀␈ε(␈α
∂␈↓ πY␈ε/for␈↓ λ⊃␈ε"0␈ε6␈α
∀␈ε(␈α
a␈ε"␈α
<␈ε(␈α
b␈ε"␈α↓.
␈β⊂6␈↓ β{␈ε+a␈↓ ∧7␈ε+t␈↓ ¬@␈ε+t
␈β⊂7␈↓ β4␈ε↓␈␈↓ ε?␈ε↓␈
␈β⊂>␈↓ βD␈∧⊂>βDα ␈↓ ε)␈∧⊂>ε)α∩
␈β⊂@␈↓ ∧B␈ε'1␈↓ ¬L␈ε-m
␈β⊂D␈↓ βD␈ε(m␈↓ ε*␈ε(b
␈β⊃⊗␈↓ ↓H␈ε/Here␈↓ α≤␈ε(↔␈↓ α@␈ε"(␈↓ αL␈ε(x␈↓ αq␈ε",␈↓ β↓␈ε".␈αε.␈αε.␈↓ β1␈ε",␈↓ βA␈ε(x␈↓ βq␈ε")␈ε/␈αis␈αthe␈αn␈α␈um␈α␈ber␈αof␈ε(␈αa␈ε/'s␈αin␈αthe␈αsequence␈↓ λJ␈ε(x␈↓ λo␈ε/,␈↓ 	¬␈ε".␈αε.␈αε.␈↓ 	5␈ε/,␈↓ 	K␈ε(x␈↓ 	{␈ε/.
␈β⊃#␈↓ α.␈ε+a␈↓ α`␈ε%1␈↓ βU␈ε+m␈↓ λ↑␈ε%1␈↓ 	←␈ε+m
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"163
␈β↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈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$␈↓ ↓H␈ε"the␈αalgorithms␈αin␈ε2␈αA␈ε"␈αcan␈αgro␈α␈w␈αas␈↓ ¬M␈ε(N␈↓ ¬|␈ε"gro␈α␈ws.
␈ββO␈↓ α␈ε"As␈αan␈αexample␈αof␈αDe|nition␈αQ2,␈αlet␈αus␈αconsider␈αbinary␈αsequences,␈αand␈αlet␈ε2␈αA
␈ββz␈↓ ↓H␈ε"be␈αjust␈αthe␈αfollo␈α␈wing␈αfour␈αalgorithms:
␈β∧9␈↓ ↓b␈ε"a)␈↓ α␈ε"T␈α⎇ak␈α␈e␈αthe␈αwh␈α↓ole␈αsequence.
␈β∧d␈↓ ↓`␈ε"b)␈↓ α␈ε"T␈α⎇ak␈α␈e␈αalternate␈αterms␈αof␈αthe␈αsequence,␈αstarting␈αwith␈αthe␈α|rst.
␈β¬⊂␈↓ ↓d␈ε"c)␈↓ α␈ε"T␈α⎇ak␈α␈e␈αthe␈αterms␈αof␈αthe␈αsequence␈αfollo␈α␈wing␈αa␈αzero.
␈β¬;␈↓ ↓`␈ε"d)␈↓ α␈ε"T␈α⎇ak␈α␈e␈αthe␈αterms␈αof␈αthe␈αsequence␈αfollo␈α␈wing␈αa␈αone.
␈β¬u␈↓ ε-␈ε%1
␈β¬z␈↓ α␈ε"No␈α␈w␈αa␈αsequence␈↓ ∧⊃␈ε(X␈↓ ∧@␈ε",␈↓ ∧V␈ε".␈αε.␈αε.␈↓ ¬ε␈ε",␈↓ ¬≤␈ε(X␈↓ ¬W␈ε"is␈α(4,␈↓ ε@␈ε")-random␈αif:
␈βεε␈↓ ∧/␈ε%1␈↓ ¬:␈ε%8
␈βε␈↓ ε-␈ε%8
␈βε∞␈↓ ε-␈∧ε∞ε-α⊂
␈βε4␈↓ α↑␈ε%1␈↓ ¬1␈ε%1␈↓ ε
␈ε%1
␈βε9␈↓ ↓H␈ε"by␈α∃(a),␈↓ αP␈ε6j␈↓ αr␈ε"(␈↓ α}␈ε(X␈↓ β4␈ε"+␈↓ β`␈ε6↓␈αε↓␈αε↓␈↓ ∧∩␈ε"+␈↓ ∧>␈ε(X␈↓ ∧m␈ε")␈ε6␈αλ␈␈↓ ¬D␈ε6j␈α
∀␈↓ ε≡␈ε",␈αi.e.,␈αif␈αthere␈αare␈α3,␈α4,␈αor␈α5␈αones;
␈βεE␈↓ β≤␈ε%1␈↓ ∧\␈ε%8
␈βεJ␈↓ α↑␈ε%8␈↓ ¬1␈ε%2␈↓ ε
␈ε%8
␈βεM␈↓ α↑␈∧εMα↑α⊂␈↓ ¬1␈∧εM¬1α⊂␈↓ ε
␈∧εMε
α⊂
␈βεi␈↓ α↑␈ε%1␈↓ ε≡␈ε%1␈↓ εz␈ε%1
␈βεn␈↓ ↓H␈ε"by␈α∀(b),␈↓ αP␈ε6j␈↓ αr␈ε"(␈↓ α}␈ε(X␈↓ β5␈ε"+␈↓ βb␈ε(X␈↓ ∧→␈ε"+␈↓ ∧F␈ε(X␈↓ ∧⎇␈ε"+␈↓ ¬*␈ε(X␈↓ ¬X␈ε")␈ε6␈α	␈␈↓ ε1␈ε6j␈α∀␈↓ π
␈ε",␈α
i.e.,␈α
if␈α
there␈α
are␈α
t␈α␈w␈α␈o␈α
ones␈α
in␈αodd-
␈βεz␈↓ β≤␈ε%1␈↓ ∧␈ε%3␈↓ ∧d␈ε%5␈↓ ¬H␈ε%7
␈βε␈␈↓ α↑␈ε%4␈↓ ε≡␈ε%2␈↓ εz␈ε%8
␈βπα␈↓ α↑␈∧παα↑α⊂␈↓ ε≡␈∧παε≡α⊂␈↓ εz␈∧παεzα⊂
␈βπ→␈↓ αP␈ε"n␈α␈um␈α␈bered␈αpositions;
␈βπN␈↓ ↓H␈ε"by␈α⊗(c),␈↓ αP␈ε"there␈αare␈αthree␈αpossibilities␈αdepending␈αon␈αh␈α↓o␈α␈w␈αman␈α␈y␈αzeros␈αoccupy␈αposi-
␈βπy␈↓ αP␈ε"tions␈↓ β*␈ε(X␈↓ βY␈ε",␈↓ βq␈ε".␈αε.␈αε.␈↓ ∧!␈ε",␈↓ ∧9␈ε(X␈↓ ∧h␈ε":␈α∂if␈α∞there␈α
are␈α∞2␈α∞or␈α
3␈α∞zeros␈α
here,␈α∂there␈α
is␈α∞n␈α↓o␈α
condition
␈βλε␈↓ βH␈ε%1␈↓ ∧W␈ε%7
␈βλ$␈↓ 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∞␈␈↓ ε3␈ε'2
␈β∂∧␈↓ εβ␈ε%2␈ε+n␈↓ ε&␈ε+∂␈↓ εA␈ε%(␈α↓1␈ε9␈␈ε+␈α␈∂␈ε%␈α↓)
␈β∂␈↓ ¬s␈ε(e␈↓ π⊃␈ε".␈↓ 
p␈ε"(35)
␈β∂ ␈↓ ¬]␈∧∂ ¬]α∩
␈β∂&␈↓ ¬]␈ε"2
␈β∂i␈↓ ↓H␈ε"This␈αresult␈α
is␈αn␈α↓ot␈αnearly␈αstrong␈αen␈α↓ough␈αto␈αsh␈α↓o␈α␈w␈αthat␈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↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈βα"␈↓ ↓H␈ε"|nite␈ε(␈α	b␈ε"-ary␈α
sequence␈↓ ∧
␈ε(X␈↓ ∧9␈ε",␈↓ ∧M␈ε".␈αε.␈αε.␈↓ ∧⎇␈ε",␈↓ ¬⊃␈ε(X␈↓ ¬L␈ε",␈α
let␈↓ ε⊃␈ε(l␈↓ ε≥␈ε"(␈↓ ε)␈ε(X␈↓ εX␈ε",␈↓ εh␈ε".␈αε.␈αε.␈↓ π_␈ε",␈↓ π(␈ε(X␈↓ πc␈ε")␈α	be␈α
the␈α	length␈α
of␈α	the␈α	sh␈α↓ortest
␈βα/␈↓ ∧(␈ε%1␈↓ ¬/␈ε+N␈↓ εG␈ε%1␈↓ πF␈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␈↓ 	π␈ε(l␈↓ 	∪␈ε"(␈↓ 	∨␈ε(X␈↓ 	N␈ε",␈↓ 	↑␈ε".␈αε.␈αε.␈↓ 
∞␈ε",␈↓ 
≡␈ε(X␈↓ 
Y␈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␈α
randomness.␈α⊂The␈α
sequences␈αof␈α
length␈↓ εh␈ε(N␈↓ π∃␈ε"that␈αmaximize␈↓ 	α␈ε(l␈↓ 	∞␈ε"(␈↓ 	~␈ε(X␈↓ 	I␈ε",␈↓ 	Y␈ε".␈αε.␈αε.␈↓ 
	␈ε",␈↓ 
→␈ε(X␈↓ 
T␈ε")␈α
ma␈α␈y
␈β∧π␈↓ 	8␈ε%1␈↓ 
7␈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#W␈α⎇e␈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␈α	R1␈α	is␈αλtherefore␈α	an␈α	appropriate␈αλbasis␈α	for␈α	theoretical␈αλstudies
␈β:␈↓ ↓H␈ε"of␈αrandomness.
␈βe␈↓ α␈ε"The␈αconcept␈α
of␈αan␈ε6␈α1␈ε"-distributed␈ε(␈αb␈ε"␈α↓-ary␈αsequence␈αw␈α␈as␈α
in␈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␈↓ 	C␈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↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈βα$␈↓ α␈ε"W␈α⎇e␈α	ha␈α␈v␈α␈e␈αλseen,␈α	h␈α↓o␈α␈w␈α␈ev␈α␈er,␈α	that␈ε6␈α	1␈ε"-distributed␈αλsequences␈α	need␈αλn␈α↓ot␈α	be␈αλsu}cien␈α␈tly
␈βαO␈↓ ↓H␈ε"haphazard␈α∞to␈α∞qualify␈α
completely␈α∞as␈α∞\random."␈α⊗Three␈α∞de|nitions,␈α∂R4,␈α∞R5,␈α∞and
␈βαz␈↓ ↓H␈ε"R6,␈α
w␈α␈ere␈αform␈α␈ulated␈α
abo␈α␈v␈α␈e␈α
to␈αpro␈α␈vide␈α
the␈α
additional␈α
conditions;␈α
and␈αDe|nition
␈ββ&␈↓ ↓H␈ε"R6,␈α∂in␈α∞particular,␈α∂seems␈α∞to␈α∞be␈α∂an␈α∞appropriate␈α∞w␈α␈a␈α␈y␈α∂to␈α∞de|ne␈α∞the␈α∂concept␈α∞of␈α∞an
␈ββQ␈↓ ↓H␈ε"in|nite␈α
random␈α∞sequence.␈α∀It␈α∞is␈α
a␈α∞precise,␈α
quan␈α␈titativ␈α␈e␈α∞statemen␈α␈t␈α
that␈α∞ma␈α␈y␈α
w␈α␈ell
␈ββ|␈↓ ↓H␈ε"coincide␈αwith␈αthe␈αin␈α␈tuitiv␈α␈e␈αidea␈αof␈αtrue␈αrandomness.
␈β∧)␈↓ α␈ε"Historically,␈α∂the␈α∂dev␈α␈elopmen␈α␈t␈α∞of␈α∂these␈α∞de|nitions␈α∞w␈α␈as␈α∂primarily␈α∞in⎇uenced
␈β∧T␈↓ ↓H␈ε"by␈αλR.␈↓ α&␈ε"v␈α␈on␈α	Mises'␈α	quest␈α	for␈α	a␈αλgo␈α↓od␈α	de|nition␈α	of␈α	\probabilit␈α␈y."␈α∂In␈ε/␈α	Math.␈αλZeitschri$
␈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␈↓ β`␈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␈α
giv␈α␈en␈αprobabilit␈α␈y␈ε(␈α
p␈ε".␈α⊃Th␈α␈us␈α
Copeland␈αw␈α␈as␈αessen␈α␈tially␈α
suggesting␈αa␈α
return␈αto
␈βπY␈↓ ↓H␈ε"De|nition␈α
R1.␈α∩Then␈α
Abraham␈↓ ¬7␈ε"W␈α⎇ald␈α
sh␈α↓o␈α␈w␈α␈ed␈α
that␈α
it␈α
is␈α
n␈α↓ot␈α
necessary␈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∃W␈α⎇ald␈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␈↓ εw␈ε"r␈↓ π¬␈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∂W␈α⎇e␈α
ha␈α␈v␈α␈e␈αobserv␈α␈ed␈α
that␈α
n␈α↓o␈αsequence␈α
can␈α
satisfy
␈β
π␈↓ ↓H␈ε"this␈αpropert␈α␈y.
␈β
4␈↓ α␈ε"The␈α	concept␈α	of␈αλ\␈↓ ∧␈ε"computabilit␈α␈y"␈α	w␈α␈as␈α	still␈α	v␈α␈ery␈α	m␈α␈uch␈αλin␈α	its␈α	infancy␈α	when␈αλW␈α⎇ald
␈β
←␈↓ ↓H␈ε"wrote␈α⊂his␈α⊂paper,␈α∩and␈α⊂A.␈↓ ∧↑␈ε"Ch␈α␈urch␈α⊃[␈ε/Bull.␈α⊂Amer.␈α⊂Math.␈α⊂Soc.␈ε2␈α⊃47␈ε"␈α⊂(1940),␈α⊃130↑135]
␈β
␈↓ ↓H␈ε"sh␈α↓o␈α␈w␈α␈ed␈α
h␈α↓o␈α␈w␈α
the␈αprecise␈α
n␈α↓otion␈α
of␈α\␈↓ ¬h␈ε"e{ectiv␈α␈e␈α
algorithm"␈α
could␈α
be␈αadded␈α
to␈α
W␈α⎇ald's
␈β5␈↓ ↓H␈ε"theory␈αto␈α
mak␈α␈e␈αhis␈α
de|nitions␈αcompletely␈α
rigorous.␈α⊃The␈αextension␈α
to␈αDe|nition
␈β`␈↓ ↓H␈ε"R6␈α
w␈α␈as␈αdue␈α
essen␈α␈tially␈α
to␈α
A.␈α
N.␈↓ ¬K␈ε"Kolm␈α↓ogoro␈α␈v␈α
[␈ε/Sankh␈α␈y␈↓ λ∃␈ε/∩␈↓ λ∃␈ε/a␈↓ λ4␈ε"(A)␈ε2␈α25␈ε"␈α
(1963),␈α
369↑376],
␈β␈↓ ↓H␈ε"wh␈α↓o␈α∞proposed␈α∂De|nition␈α∂Q2␈α∂for␈α∂|nite␈α∂sequences␈α∞in␈α∂that␈α∂same␈α∂paper.␈α_An␈α↓other
␈β7␈↓ ↓H␈ε"de|nition␈α
of␈α
randomness␈α∞for␈α
|nite␈α
sequences,␈α∞somewhere␈α
\bet␈α␈w␈α␈een"␈α
De|nitions
␈βb␈↓ ↓H␈ε"Q1␈αand␈αQ2,␈αhad␈αbeen␈αform␈α␈ulated␈αman␈α␈y␈αy␈α␈ears␈αearlier␈αby␈αA.␈αS.␈↓ 		␈ε"Besico␈α␈vitch␈α[␈ε/Math.
␈β

␈↓ ↓H␈ε/Zeitschri$␈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␈↓ ∧g␈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↑W␈α⎇ald↑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␈α	made␈α	by␈αλDonald␈α	W.␈↓ 	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↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈βα≥␈↓ εc␈ε%1␈↓ 	∞␈ε%1␈↓ 	y␈ε%1
␈βα!␈↓ ↓H␈ε"to␈α⊂Algorithm␈α⊃W␈↓ βM␈ε",␈α∩such␈α⊃that␈↓ ¬∩␈∧α!¬∩α'␈↓ ¬∩␈ε"P␈↓ ¬*␈ε"r␈↓ ¬9␈ε"(␈↓ ¬E␈ε(U␈↓ ε*␈ε6∃␈↓ εw␈ε")␈ε6␈α␈␈↓ π=␈ε"P␈↓ πU␈ε"r␈↓ πc␈ε"(␈↓ πo␈ε(U␈↓ λT␈ε6∃␈↓ 	!␈ε")␈ε6␈α∩∃␈↓ 
≥␈ε"for␈α⊂ev␈α␈ery
␈βα.␈↓ ¬↑␈ε+f␈↓ ¬o␈ε%(␈ε+n␈ε%)␈↓ λλ␈ε+f␈↓ λ→␈ε%(␈ε+n␈ε%)
␈βα2␈↓ εc␈ε%2␈↓ 	∞␈ε%2␈↓ 	y␈ε%2
␈βα5␈↓ εc␈∧α5εcα⊂␈↓ 	∞␈∧α5	∞α⊂␈↓ 	y␈∧α5	yα⊂
␈βαC␈↓ π=␈∧αCπ=α'
␈βαF␈↓ π}␈ε90
␈βαL␈↓ ↓H␈ε"R5-random␈α
sequence␈ε6␈α
h␈↓ ∧"␈ε(U␈↓ ∧O␈ε6i␈ε"␈α
produced␈α
by␈α
Algorithm␈α
W␈↓ λ∩␈ε"when␈α
it␈α
is␈α
giv␈α␈en␈α
an␈α
in|nite
␈βαY␈↓ ∧;␈ε+n
␈βαx␈↓ ↓H␈ε"set␈αof␈αsubsequence␈αrules␈↓ ∧J␈ε6R␈↓ ∧x␈ε".
␈ββ∧␈↓ ∧f␈ε+k
␈ββ#␈↓ α␈ε"Alth␈α↓ough␈α	De|nition␈αλR6␈α	is␈α	in␈α␈tuitiv␈α␈ely␈α	m␈α␈uch␈αλstronger␈α	than␈α	R4,␈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␈αbeen␈α
explored␈αby␈αD.␈α
W.␈αLo␈α␈v␈α␈eland,␈ε/␈αT␈α⎇rans.␈α
Amer.␈αMath.␈α
Soc.␈ε2␈α125␈ε"␈α
(1966),
␈βλV␈↓ ↓H␈ε"497↑510.␈α∨See␈α⊃also␈α∩C.-P.␈↓ ∧e␈ε"Schn␈α↓orr␈α⊃[␈ε/Zeitschr.␈α⊃W␈α⎇ahr.␈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.␈↓ ε0␈ε"Brou␈α␈w␈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␈↓ αY␈ε/∪␈↓ αY␈ε/a␈↓ αk␈ε/lligk␈α␈eit␈α∞und␈α∞W␈α⎇ahrscheinlichk␈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␈↓ ∧H␈ε:b␈↓ ∧S␈ε&lg␈↓ ∧i␈ε&(␈ε,n␈ε&+1)␈ε:␈α␈c
␈β
l␈↓ ↓;␈ε↓x␈↓ λn␈ε&1
␈β
n␈↓ ↓c␈ε34.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈22␈↓ αt␈ε#]␈α⊗Let␈↓ βP␈ε)U␈↓ ∧α␈ε#=␈α	(␈↓ ∧8␈ε#2␈↓ ¬B␈ε#/␈α␈3)␈↓ ¬t␈ε#mo␈α␈d␈↓ ε8␈ε#1.␈α∂Wha␈α␈t␈αi␈α↓s␈↓ π←␈ε#P␈↓ πu␈ε#r␈↓ λα␈ε#(␈↓ λ
␈ε)U␈↓ λ?␈ε#<␈↓ λ␈␈ε#)?
␈β
z␈↓ βf␈ε,n␈↓ λ$␈ε,n
␈β
␈␈↓ λn␈ε&2
␈β∞↓␈↓ λn␈∧∞↓λnα∞
␈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∂@␈↓ ↓S␈ε#P␈↓ ↓i␈ε#r␈↓ ↓v␈ε#(␈↓ α↓␈ε)u␈↓ α-␈ε7∀␈↓ αX␈ε)U␈↓ β
␈ε#<␈↓ β5␈ε)v␈↓ β←␈ε#or␈↓ ∧λ␈ε)u␈↓ ∧4␈ε7∀␈↓ ∧↑␈ε)U␈↓ ¬9␈ε#<␈↓ ¬c␈ε)v␈↓ εβ␈ε#)␈α	=␈↓ εB␈ε)v␈↓ εi␈ε7␈␈↓ π∩␈ε)u␈↓ π<␈ε#+␈↓ πd␈ε)v␈↓ λ␈ε7␈␈↓ λ4␈ε)u␈↓ λ↑␈ε7␈␈ε#␈αλ(␈↓ 	∩␈ε)v␈↓ 	9␈ε7␈␈↓ 	a␈ε)u␈↓ 
∧␈ε#)(␈↓ 
~␈ε)v␈↓ 
A␈ε7␈␈↓ 
j␈ε)u␈↓ 
␈ε#).
␈β∂K␈↓ α∃␈ε&1␈↓ αo␈ε,n␈↓ βE␈ε&1␈↓ ∧≠␈ε&2␈↓ ∧u␈ε,n␈ε&+1␈↓ ¬t␈ε&2␈↓ εR␈ε&1␈↓ π%␈ε&1␈↓ πu␈ε&2␈↓ λG␈ε&2␈↓ 	"␈ε&1␈↓ 	u␈ε&1␈↓ 
+␈ε&2␈↓ 
⎇␈ε&2
␈β⊂≠␈↓ ↓g␈ε36.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈23␈↓ αt␈ε#]␈α⊗Let␈↓ βM␈ε)S␈↓ βp␈ε#(␈ε)n␈ε#␈α↓),␈↓ ∧-␈ε)S␈↓ ∧P␈ε#(␈ε)n␈ε#␈α↓),␈↓ ¬
␈ε#.␈αε.␈αε.␈↓ ¬<␈ε#be␈αλan␈αλin␈α␈|n␈α␈i␈α↓te␈αλseq␈α␈uen␈α␈ce␈αλof␈αλstatem␈α␈en␈α␈ts␈αλab␈α␈out␈αλm␈α␈u␈α␈tua␈α␈l␈α↓ly
␈β⊂&␈↓ βa␈ε&1␈↓ ∧A␈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
an␈α␈d␈↓ ¬
␈ε)S␈↓ ¬,␈ε#(␈ε)n␈ε#)␈α
ca␈α␈nnot␈α
sim␈α␈u␈α␈ltan␈α␈eou␈α␈sl␈α↓y␈αb␈α␈e␈α
tru␈α␈e␈α
if␈ε)␈α
i␈ε7␈α≤␈↓ 
⊃␈ε)j␈↓ 
!␈ε#.␈α∪Assume
␈β⊂N␈↓ ∧ε␈ε,i␈↓ ¬≡␈ε,j
␈β⊂h␈↓ α5␈ε"(␈↓ β
␈ε")␈↓ πG␈ε"(␈↓ 
s␈ε")
␈β⊂j␈↓ ↓H␈ε#th␈α␈at␈↓ α∩␈ε#P␈↓ α(␈ε#r␈↓ αA␈ε)S␈↓ αc␈ε#(␈ε)n␈ε#)␈↓ β&␈ε#ex␈α␈i␈α↓st␈α␈s␈α
for␈α
ea␈α␈ch␈↓ ¬λ␈ε)j␈↓ ¬#␈ε7∃␈ε#␈α
1.␈α∀Sho␈α␈w␈α
th␈α␈at␈↓ π$␈ε#P␈↓ π:␈ε#r␈↓ πS␈ε)S␈↓ πu␈ε#(␈ε)n␈ε#)␈↓ λ*␈ε#is␈αtrue␈αfor␈αso␈α␈me␈↓ 
→␈ε)j␈↓ 
5␈ε7∃␈ε#␈α1␈↓ ␈ε7∃
␈β⊂u␈↓ ↓H␈ε↓P␈↓ αU␈ε,j␈↓ πg␈ε,j
␈β⊃	␈↓ π$␈∧⊃	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␈↓ ↓z␈ε:∃␈ε&1
␈β∪(

␈β↓U␈↓ ↓H␈ε"3.5␈↓ 
v␈ε"167
␈β↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈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␈↓ ↓|␈ε7∃␈ε#␈α	1.␈α⊂Assu␈α␈me␈αtha␈α␈t␈αfo␈α␈r␈αa␈α␈l␈α↓l␈ε)␈αn␈ε#␈α
>␈α
0,␈↓ ¬c␈ε)S␈↓ ε∂␈ε#(␈ε)n␈ε#)␈αis␈αtrue␈αfor␈αex␈α␈act␈α␈l␈α↓y␈αo␈α␈ne␈αp␈α␈air␈αo␈α␈f␈αin␈α␈t␈α␈egers␈ε)␈αi␈ε#,␈↓ ∀␈ε)j␈↓ #␈ε#.
␈βαW␈↓ ↓g␈ε↓P␈↓ ¬w␈ε,i␈↓ ε↓␈ε,j
␈βαq␈↓ α}␈ε"(␈↓ β`␈ε")␈↓ π¬␈ε"(␈↓ 
5␈ε")
␈βαs␈↓ ↓H␈ε#If␈↓ α[␈ε#P␈↓ αq␈ε#r␈↓ β
␈ε)S␈↓ β6␈ε#(␈ε)n␈ε#)␈↓ βu␈ε#=␈α
1,␈α
d␈α␈oe␈α␈s␈α
it␈α	foll␈α↓o␈α}w␈α
th␈α␈at␈α	\␈↓ εb␈ε#P␈↓ εx␈ε#r␈↓ π⊃␈ε)S␈↓ π=␈ε#(␈ε)n␈ε#)␈↓ πr␈ε#i␈α↓s␈αtr␈α␈ue␈αfor␈αso␈α␈me␈↓ 	a␈ε)j␈↓ 	z␈ε7∃␈ε#␈α	1␈↓ 
A␈ε#"␈α	ex␈α␈i␈α↓st␈α␈s
␈βα␈␈↓ β≡␈ε,i␈↓ β(␈ε,j␈↓ π%␈ε,i␈↓ π/␈ε,j
␈ββε␈↓ ¬≤␈ε↓P
␈ββπ␈↓ α
␈ε,i␈ε&,␈↓ α∨␈ε,j␈↓ α,␈ε:∃␈ε&1
␈ββ ␈↓ ε!␈ε"(␈↓ πβ␈ε")
␈ββ"␈↓ ↓H␈ε#fo␈α␈r␈αall␈ε)␈αi␈ε7␈α	∃␈ε#␈α	1,␈αan␈α␈d␈αtha␈α␈t␈αi␈α↓t␈αe␈α␈qua␈α␈ls␈↓ ¬}␈ε#P␈↓ ε∀␈ε#r␈↓ ε-␈ε)S␈↓ εY␈ε#(␈ε)n␈ε#)␈↓ π∂␈ε#?
␈ββ.␈↓ εA␈ε,i␈↓ εK␈ε,j
␈ββ6␈↓ ¬B␈ε,j␈↓ ¬O␈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#)␈↓ 	1␈ε#.␈α↓]
␈β∧∪␈↓ λ3␈ε,j␈↓ λ?␈ε,n
␈β∧≠␈↓ π/␈ε&1␈ε:∀␈↓ πW␈ε,j␈↓ πd␈ε:∀␈ε,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␈αλp␈α␈ro␈α␈v␈α␈e␈αλt␈α␈hat␈αλif␈ε7␈αλh␈↓ εJ␈ε)U␈↓ εs␈ε7i␈ε#␈αλi␈α↓s␈↓ π%␈ε71␈ε#-distrib␈α␈uted␈α␈,␈α	so␈αλis␈αλthe␈αλsu␈α␈bse␈α␈que␈α␈nce
␈β∧y␈↓ εa␈ε,n
␈β¬∃␈↓ ↓H␈ε7h␈↓ ↓S␈ε)U␈↓ α	␈ε7i␈ε#␈α↓.
␈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␈↓ 	p␈ε,k
␈β¬k␈↓ ∧⊂␈ε"(␈↓ λK␈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␈↓ ¬V␈ε#,␈↓ ¬e␈ε)U␈↓ ε6␈ε#,␈↓ εE␈ε#.␈αε.␈αε.␈↓ εq␈ε#,␈↓ π␈ε)U␈↓ π{␈ε#)␈α	<␈↓ λ9␈ε)v␈↓ λ`␈ε#=␈↓ 	␈ε)v␈↓ 	4␈ε7␈␈↓ 	]␈ε)u␈↓ 
␈ε#.
␈β¬y␈↓ ¬D␈ε,n␈↓ ¬|␈ε,n␈ε&+1␈↓ π↔␈ε,n␈ε&+␈↓ πB␈ε,k␈↓ πQ␈ε:␈␈ε&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<␈↓ ∧r␈ε)␈␈↓ ¬∩␈ε7∀␈ε#␈α1␈αand␈ε)␈αp␈ε#␈α=␈↓ εh␈ε)␈␈↓ π¬␈ε7␈␈ε)␈α	␈ε#␈α␈,␈α
l␈α↓e␈α␈t␈↓ λ␈ε)f␈↓ λ ␈ε#(0␈α␈)␈α=␈α0,␈α
an␈α␈d␈αfor␈ε)␈αn␈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␈↓ λV␈ε#<␈↓ 	↓␈ε)␈␈↓ 	⊗␈ε#;␈αthe␈α␈n
␈βεy␈↓ λ4␈ε,m
␈βπ'␈↓ λ-␈ε,k␈↓ λ;␈ε:␈␈ε&␈α↓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	=␈↓ εV␈ε)k␈↓ π∧␈ε#=␈ε)␈α∂p␈ε#(1␈ε7␈απ␈␈ε)␈αλp␈↓ 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␈↓ αq␈ε#>␈↓ β≤␈ε)U␈↓ βL␈ε#,␈αth␈α␈en
␈βλT␈↓ α&␈ε,m␈ε:␈␈ε&␈α↓1␈↓ β2␈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	the␈α␈re␈α
a␈α␈re␈α
o␈α␈nly␈α	t␈α␈w␈α␈o␈α
k␈α␈ind␈α␈s␈α
of␈α	cou␈α␈po␈α␈ns,␈α
in␈α	the␈α	follo␈α␈wing␈α	sen␈α␈se:␈αLet␈↓ 	2␈ε)X␈↓ 	]␈ε#,␈↓ 	p␈ε)X␈↓ 
≠␈ε#,␈↓ 
.␈ε#.␈αε.␈αε.␈↓ 
↑␈ε#be␈α	an
␈β
λ␈↓ 	N␈ε&1␈↓ 
␈ε&2
␈β
$␈↓ ↓H␈ε71␈ε#-distrib␈α␈uted␈αbin␈α␈ary␈αse␈α␈que␈α␈nce␈α␈.␈α∩Let␈↓ ¬←␈ε)f␈↓ ¬s␈ε#(0)␈α=␈α0␈α␈,␈α
a␈α␈nd␈αf␈α↓o␈α␈r␈ε)␈αn␈ε7␈α∃␈ε#␈α1␈αlet␈↓ 	ε␈ε)f␈↓ 	~␈ε#(␈ε)n␈ε#)␈αbe␈αth␈α␈e␈αsma␈α␈ll␈α↓es␈α␈t
␈β
K␈↓ ↓H␈ε#in␈α}teger␈ε)␈α
m␈ε#␈α
>␈↓ β∩␈ε)f␈↓ β%␈ε#(␈ε)␈α↓n␈ε7␈α	␈␈ε#␈α	1)␈α∞su␈α␈ch␈α
th␈α␈at␈ε7␈α∞f␈↓ ¬K␈ε)X␈↓ εl␈ε#,␈↓ ε{␈ε#.␈αε.␈α¬.␈↓ π'␈ε#,␈↓ π6␈ε)X␈↓ πk␈ε7g␈ε#␈α
is␈α∞th␈α␈e␈α
set␈ε7␈α∞f␈ε#␈α␈0,␈αε1␈ε7␈α␈g␈ε#.␈α↔Pro␈α}v␈α␈e␈α
tha␈α␈t
␈β
W␈↓ πQ␈ε,m
␈β
X␈↓ ¬g␈ε,f␈↓ ¬v␈ε&(␈ε,n␈ε:␈␈ε&1)␈α␈+␈α↓1
␈β
m␈↓ ¬ε␈ε&1␈ε:␈␈↓ ¬.␈ε,k
␈β
q␈↓ ↓k␈ε"(␈↓ ∧+␈ε")
␈β
s␈↓ ↓H␈ε#P␈↓ ↓↑␈ε#r␈↓ ↓w␈ε)f␈↓ α␈ε#(␈ε)n␈ε#)␈ε7␈αλ␈␈↓ αe␈ε)f␈↓ αy␈ε#(␈ε)n␈ε7␈αλ␈␈ε#␈αλ1␈α␈)␈α
=␈↓ ∧_␈ε)k␈↓ ∧F␈ε#=␈↓ ∧v␈ε#2␈↓ ¬?␈ε#,␈αfor␈↓ επ␈ε)k␈↓ ε#␈ε7∃␈ε#␈α	2.␈α→(Cf.␈αex␈α␈ercise␈α7␈α␈.␈α↓)
␈β$␈↓ ↓V␈ε316.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈38␈↓ αt␈ε#]␈α⊗Do␈α␈es␈α
th␈α␈e␈αcou␈α␈po␈α␈n-collecto␈α␈r's␈α
test␈αhold␈αfor␈ε7␈α1␈ε#␈α↓-d␈α␈i␈α↓st␈α␈ri␈α↓b␈α␈ute␈α␈d␈αsequ␈α␈enc␈α␈es␈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␈↓ ↓z␈ε#mo␈α␈d␈↓ α>␈ε#1␈ε7i␈ε#␈α⊂is␈α⊂not␈α⊂2-d␈α␈i␈α↓s␈α␈tri␈α↓b␈α␈ute␈α␈d.␈α≥B␈α↓u␈α␈t␈α⊂is␈α⊂ther␈α␈e␈α⊂an␈α}y␈α⊂ratio␈α␈nal␈α⊂n␈α}um␈α␈b␈α␈er␈↓ 	:␈ε)r␈↓ 	Y␈ε#for␈α⊂wh␈α␈i␈α↓c␈α␈h␈α⊂th␈α␈is
␈βI␈↓ ∞␈ε&3
␈βK␈↓ ↓H␈ε#se␈α␈que␈α␈nce␈α	is␈α
equ␈α␈idistribu␈α␈ted?␈α∞In␈α	pa␈α␈rticular,␈α
is␈α
th␈α␈e␈α
seq␈α␈uen␈α␈ce␈α
e␈α␈qu␈α␈i␈α↓d␈α␈istri␈α↓b␈α␈ute␈α␈d␈α
wh␈α␈en␈↓ 
G␈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(19␈α␈57),␈α12␈α␈2↑1␈α␈24.]
␈β
"␈↓ ↓;␈ε↓x
␈β
$␈↓ ↓S␈ε31␈α␈8.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈22␈↓ αt␈ε#]␈α⊗Pro␈α␈v␈α}e␈αth␈α␈at␈αif␈↓ ∧Z␈ε)U␈↓ ¬␈ε#,␈↓ ¬∃␈ε)U␈↓ ¬;␈ε#,␈↓ ¬P␈ε#.␈αε.␈αε.␈↓ εβ␈ε#is␈↓ ε%␈ε)k␈↓ ε7␈ε#-distribu␈α␈ted␈α␈,␈α
so␈αis␈αth␈α␈e␈αseq␈α␈uen␈α␈ce␈↓ 
ε␈ε)V␈↓ 
)␈ε#,␈↓ 
>␈ε)V␈↓ 
b␈ε#,␈↓ 
w␈ε#.␈αε.␈α¬.␈↓ #␈ε#,
␈β
/␈↓ ∧q␈ε&0␈↓ ¬,␈ε&1␈↓ 
~␈ε&0␈↓ 
R␈ε&1
␈β
K␈↓ ↓H␈ε#wh␈α␈ere␈↓ α(␈ε)V␈↓ αW␈ε#=␈ε7␈α
b␈ε)n␈↓ β#␈ε)U␈↓ βL␈ε7c␈ε#/␈ε)n␈ε#.
␈β
W␈↓ α<␈ε,n␈↓ β:␈ε,n
␈β
|␈↓ ↓V␈ε319.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈35␈↓ αt␈ε#]␈α⊗Co␈α␈nside␈α␈r␈αa␈αm␈α↓o␈α␈di|c␈α␈ation␈αof␈αDe|␈α␈nition␈αR4␈αtha␈α␈t␈αre␈α␈quires␈αth␈α␈e␈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∂≡␈↓ ¬M␈ε,n
␈β∂#␈↓ ↓V␈ε320.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈47␈↓ αt␈ε#]␈α⊗Do␈α␈es␈α⊂th␈α␈e␈α⊂seq␈α␈uen␈α␈ce␈ε7␈α∂h␈↓ ¬>␈ε)∩␈↓ ¬e␈ε#mo␈α␈d␈↓ ε)␈ε#1␈ε7i␈ε#␈α⊂satisfy␈α∂De|␈α␈nition␈α∂R4␈α⊂fo␈α␈r␈α⊂alm␈α↓o␈α␈st␈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␈↓ βr␈ε#b␈α␈e␈αa␈α
set␈αan␈α␈d␈αlet␈ε7␈α
M␈ε#␈αb␈α␈e␈αa␈αc␈α␈oll␈α↓e␈α␈ction␈α
of␈αsu␈α␈bsets␈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␈α
i␈α↓n␈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	\ran␈α␈do␈α␈mly"␈α
c␈α␈h␈α↓o␈α␈sen␈α
elem␈α␈en␈α␈t␈α
o␈α␈f␈↓ ¬"␈ε)S␈↓ ¬D␈ε#lies␈α
in␈↓ ε"␈ε)M␈↓ εG␈ε#.␈α∂Gen␈α␈eralize␈α
De|␈α␈nition␈α␈s␈α
B␈α
and␈α	D␈α
to␈α
ob␈α␈tain
␈β⊂r␈↓ ↓H␈ε#a␈α
g␈α␈o␈α↓od␈α	de|␈α␈nition␈α
o␈α␈f␈αth␈α␈e␈α
con␈α␈cep␈α␈t␈αo␈α␈f␈αa␈↓ ¬←␈ε)k␈↓ ¬q␈ε#-␈α↓d␈α␈istribu␈α␈ted␈α
seq␈α␈uen␈α␈ce␈ε7␈α
h␈↓ λ9␈ε)Z␈↓ λa␈ε7i␈ε#␈αo␈α␈f␈αelem␈α␈en␈α␈ts␈α
of␈↓ 
J␈ε)S␈↓ 
l␈ε#with
␈β⊂}␈↓ λO␈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↓\␈↓ α=␈ε∞RANDOM␈α
NU␈α↓MB␈α␈E␈α↓RS
␈βα#␈↓ ↓;␈ε↓x
␈βα%␈↓ ↓S␈ε32␈α␈2.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈30␈↓ αt␈ε#]␈α⊗(Herm␈α␈ann␈↓ ∧/␈ε#W␈α⎇eyl.)␈α≠Sho␈α␈w␈αth␈α␈at␈αt␈α␈he␈↓ πβ␈ε#[␈↓ π∩␈ε#0␈α␈,␈αε1)␈αseq␈α␈uen␈α␈ce␈↓ λc␈ε7h␈↓ λn␈ε)U␈↓ 	⊗␈ε7i␈ε#␈αi␈α↓s␈↓ 	P␈ε)k␈↓ 	b␈ε#-distrib␈α␈uted␈αif
␈βα1␈↓ 	∧␈ε,n
␈βαM␈↓ ↓H␈ε#a␈α␈nd␈α
only␈αif
␈βββ␈↓ ∧?␈ε↓X
␈ββ⊃␈↓ ∧↓␈ε#1
␈ββ%␈↓ ¬I␈ε"(␈↓ 	α␈ε")
␈ββ'␈↓ β2␈ε#lim␈↓ ¬∃␈ε#e␈↓ ¬$␈ε#x␈↓ ¬7␈ε#p␈↓ ¬U␈ε#2␈↓ ¬f␈ε)→␈↓ ¬z␈ε)i␈ε#(␈↓ ε⊃␈ε)c␈↓ ε/␈ε)U␈↓ ε←␈ε#+␈↓ πλ␈ε7↓␈α¬↓␈αε↓␈↓ π6␈ε#+␈↓ π←␈ε)c␈↓ π⎇␈ε)U␈↓ λw␈ε#)␈↓ 	_␈ε#=␈α	0
␈ββ2␈↓ ε∨␈ε&1␈↓ εE␈ε,n␈↓ πm␈ε,k␈↓ λ∀␈ε,n␈ε&␈α␈+␈↓ λ?␈ε,k␈↓ λN␈ε:␈␈ε&1
␈ββ:␈↓ βy␈∧β:βyα 
␈ββ>␈↓ βy␈ε)N
␈ββH␈↓ β!␈ε,N␈↓ β:␈ε:!1
␈ββY␈↓ ∧#␈ε&0␈ε:␈α␈∀␈ε,␈α↓n␈ε&<␈↓ ∧v␈ε,N
␈β∧_␈↓ ↓H␈ε#fo␈α␈r␈αev␈α␈ery␈α
set␈αof␈αin␈α␈teg␈α␈ers␈↓ ∧)␈ε)c␈↓ ∧G␈ε#,␈↓ ∧[␈ε)c␈↓ ∧x␈ε#,␈↓ ¬
␈ε#.␈αε.␈α¬.␈↓ ¬9␈ε#,␈↓ ¬N␈ε)c␈↓ ¬w␈ε#n␈α↓o␈α␈t␈αall␈αz␈α␈ero.
␈β∧$␈↓ ∧7␈ε&1␈↓ ∧i␈ε&2␈↓ ¬\␈ε,k
␈β∧J␈↓ ↓V␈ε323.␈↓ α␈ε#[␈ε)M3␈α␈4␈↓ α\␈ε#]␈α⊗Sho␈α␈w␈αthat␈αa␈ε)␈αb␈ε#␈α↓-a␈α␈ry␈αsequ␈α␈en␈α␈ce␈ε7␈α
h␈↓ ε'␈ε)X␈↓ εT␈ε7i␈ε#␈α
is␈↓ π∂␈ε)k␈↓ π!␈ε#-distribu␈α␈ted␈αif␈α
a␈α␈nd␈αon␈α␈ly␈αi␈α↓f␈αall␈α
of␈αthe
␈β∧V␈↓ εB␈ε,n
␈β∧r␈↓ ↓H␈ε#se␈α␈que␈α␈nce␈α␈s␈ε7␈αh␈↓ αi␈ε)c␈↓ βπ␈ε)X␈↓ β=␈ε#+␈↓ βf␈ε)c␈↓ ∧∧␈ε)X␈↓ ∧a␈ε#+␈↓ ¬␈ε7↓␈αε↓␈α¬↓␈↓ ¬:␈ε#+␈↓ ¬c␈ε)c␈↓ ε↓␈ε)X␈↓ π␈ε7i␈ε#␈αare␈ε0␈αeq␈α␈uidistrib␈α␈uted␈ε#␈α␈,␈α
wh␈α␈en␈α␈ev␈α␈er␈↓ 
S␈ε)c␈↓ 
p␈ε#,␈↓ ε␈ε)c␈↓ #␈ε#,
␈β∧⎇␈↓ αx␈ε&1␈↓ β#␈ε,n␈↓ βt␈ε&2␈↓ ∧∨␈ε,n␈ε&+1␈↓ ¬q␈ε,k␈↓ ε≥␈ε,n␈ε&+␈↓ εH␈ε,k␈↓ εW␈ε:␈␈ε&1␈↓ 
a␈ε&1␈↓ ∀␈ε&2
␈β¬→␈↓ ↓H␈ε#.␈α¬.␈αε.␈↓ ↓t␈ε#,␈↓ αλ␈ε)c␈↓ α2␈ε#are␈αin␈α}teger␈α␈s␈αwith␈↓ ∧0␈ε#gcd␈↓ ∧b␈ε#(␈↓ ∧m␈ε)c␈↓ ¬␈ε#,␈↓ ¬~␈ε#.␈αε.␈α¬.␈↓ ¬F␈ε#,␈↓ ¬U␈ε)c␈↓ ¬s␈ε#)␈α
=␈α	1.
␈β¬%␈↓ α↔␈ε,k␈↓ ∧|␈ε&1␈↓ ¬c␈ε,k
␈β¬K␈↓ ↓V␈ε324.␈↓ α␈ε#[␈ε)M2␈α␈5␈↓ α\␈ε#]␈α⊗Sho␈α␈w␈α
th␈α␈at␈αa␈↓ ∧=␈ε#[␈↓ ∧L␈ε#0,␈α¬1)␈α
seq␈α␈uen␈α␈ce␈ε7␈αh␈↓ ε*␈ε)U␈↓ εR␈ε7i␈ε#␈α
is␈↓ π
␈ε)k␈↓ π ␈ε#-d␈α␈i␈α↓s␈α␈tri␈α↓b␈α␈ute␈α␈d␈α
if␈α
a␈α␈nd␈αon␈α␈ly␈α
if␈α
a␈α␈l␈α↓l␈α
o␈α␈f␈α
the
␈β¬W␈↓ ε@␈ε,n
␈β¬s␈↓ ↓H␈ε#se␈α␈que␈α␈nce␈α␈s␈ε7␈α∞h␈↓ αk␈ε)c␈↓ βλ␈ε)U␈↓ β:␈ε#+␈↓ βd␈ε)c␈↓ ∧α␈ε)U␈↓ ∧\␈ε#+␈↓ ¬ε␈ε7↓␈αε↓␈αε↓␈↓ ¬6␈ε#+␈↓ ¬`␈ε)c␈↓ ¬␈␈ε)U␈↓ εy␈ε7i␈ε#␈α
are␈ε0␈α
eq␈α␈uidistribu␈α␈ted␈ε#␈α␈,␈α∂wh␈α␈en␈α␈ev␈α␈er␈↓ 
Q␈ε)c␈↓ 
n␈ε#,␈↓ ε␈ε)c␈↓ #␈ε#,
␈β¬␈␈↓ αy␈ε&1␈↓ β∨␈ε,n␈↓ βs␈ε&2␈↓ ∧→␈ε,n␈ε&␈α␈+␈α↓1␈↓ ¬n␈ε,k␈↓ ε∃␈ε,n␈ε&+␈↓ ε@␈ε,k␈↓ εO␈ε:␈␈ε&␈α↓1␈↓ 
←␈ε&1␈↓ ∀␈ε&2
␈βε≠␈↓ ↓H␈ε#.␈α¬.␈αε.␈↓ ↓t␈ε#,␈↓ αλ␈ε)c␈↓ α2␈ε#are␈αin␈α}teger␈α␈s␈αnot␈αa␈α␈l␈α↓l␈αzero␈α␈.
␈βε&␈↓ α↔␈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␈↓ βd␈ε#1
␈βλ≠␈↓ ¬D␈ε&1␈↓ εm␈ε&1
␈βλ≥␈↓ β≡␈ε#l␈α↓im␈↓ ∧a␈ε#(␈↓ ∧l␈ε)U␈↓ ¬_␈ε7␈␈↓ ¬V␈ε#)(␈↓ ¬l␈ε)U␈↓ εA␈ε7␈␈↓ ε␈␈ε#)␈α	=␈α
0␈α␈,␈↓ λ ␈ε#for␈αa␈α␈l␈α↓l␈↓ 	α␈ε)k␈↓ 	≡␈ε7∃␈ε#␈α	1.
␈βλ)␈↓ ¬β␈ε,j␈↓ εβ␈ε,j␈↓ ε∂␈ε&+␈↓ ε)␈ε,k
␈βλ.␈↓ ¬D␈ε&2␈↓ εm␈ε&2
␈βλ0␈↓ βb␈∧λ0βbα∃␈↓ ¬D␈∧λ0¬Dα∞␈↓ εm␈∧λ0εmα∞
␈βλ4␈↓ βb␈ε)n
␈βλ?␈↓ β∩␈ε,n␈ε:!1
␈βλP␈↓ ∧␈ε&0␈ε:∀␈↓ ∧(␈ε,j␈↓ ∧4␈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␈αbe␈αra␈α␈nd␈α␈om.␈α∂Let␈↓ ¬,␈ε)U␈↓ ¬R␈ε#,␈↓ ¬f␈ε)U␈↓ ε␈ε#,␈↓ ε!␈ε#.␈αε.␈α¬.␈↓ εS␈ε#b␈α␈e␈αan␈ε7␈α1␈ε#-d␈α␈i␈α↓strib␈α␈uted␈α
sequ␈α␈en␈α␈ce,␈αa␈α␈nd␈α
de|␈α␈ne
␈β	N␈↓ ¬C␈ε&0␈↓ ¬⎇␈ε&1
␈β	j␈↓ ↓H␈ε#th␈α␈e␈αseq␈α␈uen␈α␈ce␈↓ β
␈ε)V␈↓ β-␈ε#,␈↓ βB␈ε)V␈↓ βe␈ε#,␈↓ βy␈ε#.␈αε.␈αε.␈↓ ∧+␈ε#as␈αfollo␈α␈ws:
␈β	v␈↓ β≡␈ε&0␈↓ βV␈ε&1
␈β
1␈↓ β4␈ε#(␈↓ β?␈ε)V␈↓ ∧≠␈ε#,␈↓ ∧*␈ε)V␈↓ ∧↑␈ε#)␈α
=␈α	(␈↓ ¬(␈ε)U␈↓ επ␈ε#,␈↓ ε⊗␈ε)U␈↓ εL␈ε#)␈↓ π~␈ε#if␈α(␈↓ πE␈ε)U␈↓ λ$␈ε#,␈↓ λ2␈ε)U␈↓ λi␈ε#)␈ε7␈α	2␈ε)␈α
G␈ε#,
␈β
=␈↓ βS␈ε&2␈ε,n␈ε:␈␈ε&1␈↓ ∧>␈ε&2␈ε,n␈↓ ¬?␈ε&2␈ε,n␈ε:␈α␈␈␈ε&␈α↓1␈↓ ε,␈ε&2␈ε,n␈↓ π[␈ε&2␈ε,n␈ε:␈␈ε&1␈↓ λI␈ε&2␈ε,n
␈β
Y␈↓ β4␈ε#(␈↓ β?␈ε)V␈↓ ∧≠␈ε#,␈↓ ∧*␈ε)V␈↓ ∧↑␈ε#)␈α
=␈α	(␈↓ ¬(␈ε)U␈↓ ¬←␈ε#,␈↓ ¬n␈ε)U␈↓ εL␈ε#)␈↓ π~␈ε#if␈α(␈↓ πE␈ε)U␈↓ λ$␈ε#,␈↓ λ2␈ε)U␈↓ λi␈ε#)␈ε7␈α	3␈ε)␈α
G␈ε#,
␈β
e␈↓ βS␈ε&2␈ε,n␈ε:␈␈ε&1␈↓ ∧>␈ε&2␈ε,n␈↓ ¬?␈ε&2␈ε,n␈↓ ε∧␈ε&2␈ε,n␈ε:␈␈ε&1␈↓ π[␈ε&2␈ε,n␈ε:␈␈ε&1␈↓ λI␈ε&2␈ε,n
␈β ␈↓ ¬≠␈ε&1␈↓ π2␈ε&1
␈β"␈↓ ↓H␈ε#wh␈α␈ere␈ε)␈αG␈ε#␈αis␈αth␈α␈e␈αset␈↓ β[␈ε7f␈↓ βq␈ε#(␈ε)x␈ε#␈α␈,␈↓ ∧≥␈ε)y␈↓ ∧/␈ε#)␈ε7␈α	j␈ε)␈α
x␈ε7␈αε␈␈↓ ¬6␈ε7∀␈↓ ¬`␈ε)y␈↓ ¬{␈ε7∀␈ε)␈α
x␈↓ εC␈ε#or␈↓ εl␈ε)x␈ε#␈απ+␈↓ πL␈ε7∀␈↓ πw␈ε)y␈↓ λ∞␈ε7g␈↓ λ∨␈ε#.␈α∂S␈α␈h␈α↓o␈α}w␈αtha␈α␈t␈α→(a)␈↓ 
↔␈ε)V␈↓ 
:␈ε#,␈↓ 
N␈ε)V␈↓ 
q␈ε#,␈↓ ε␈ε#.␈α¬.␈αε.
␈β.␈↓ 
+␈ε&0␈↓ 
b␈ε&1
␈β3␈↓ ¬≠␈ε&2␈↓ π2␈ε&2
␈β5␈↓ ¬≠␈∧5¬≠α∞␈↓ π2␈∧5π2α∞
␈βG␈↓ πJ␈ε&5
␈βJ␈↓ ↓H␈ε#is␈α
equ␈α␈i␈α↓d␈α␈istribu␈α␈ted␈α
an␈α␈d␈α
white;␈α↔(b)␈↓ ¬0␈ε#P␈↓ ¬G␈ε#r␈↓ ¬T␈ε#(␈↓ ¬←␈ε)V␈↓ ε∞␈ε#>␈↓ ε9␈ε)V␈↓ πλ␈ε#)␈α	=␈↓ π\␈ε#.␈α↔(This␈α
po␈α␈i␈α↓n␈α}ts␈αo␈α␈ut␈α
the␈α
w␈α␈eak␈α␈nes␈α␈s
␈βU␈↓ ¬s␈ε,n␈↓ εM␈ε,n␈ε&+1
␈βZ␈↓ πJ␈ε&8
␈β]␈↓ πJ␈∧]πJα∞
␈βq␈↓ ↓H␈ε#o␈α␈f␈αt␈α␈he␈αseria␈α␈l␈αco␈α␈rrelation␈α
test.)
␈β$␈↓ ↓V␈ε327.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈48␈↓ αt␈ε#]␈α⊗Wha␈α␈t␈αi␈α↓s␈αth␈α␈e␈αh␈α␈igh␈α␈est␈αp␈α␈ossib␈α␈l␈α↓e␈αv␈α⎇alue␈αfor␈↓ πD␈ε#P␈↓ πZ␈ε#r␈↓ πg␈ε#(␈↓ πr␈ε)V␈↓ λ"␈ε#>␈↓ λM␈ε)V␈↓ 	≤␈ε#)␈αin␈αan␈αeq␈α␈uid␈α␈i␈α↓strib␈α␈-
␈β/␈↓ λπ␈ε,n␈↓ λa␈ε,n␈ε&+␈α↓1
␈βK␈↓ ↓H␈ε#u␈α␈ted␈α␈,␈αwhite␈α
se␈α␈que␈α␈nce?␈α∃(D.␈↓ ∧M␈ε#Co␈α␈pp␈α␈ersmith␈α
h␈α␈as␈α
co␈α␈nstru␈α␈cted␈α	such␈α	a␈α
seq␈α␈uen␈α␈ce␈α
ach␈α␈ievin␈α␈g␈α
the
␈βp␈↓ α!␈ε&7
␈βs␈↓ ↓H␈ε#v␈α⎇alue␈↓ α3␈ε#.)
␈β
β␈↓ α!␈ε&8
␈β
ε␈↓ α!␈∧
εα!α∞
␈β
#␈↓ ↓;␈ε↓x
␈β
%␈↓ ↓S␈ε32␈α␈8.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈21␈↓ αt␈ε#]␈α⊗Use␈α	the␈α
se␈α␈que␈α␈nce␈α	(␈α↓1␈α␈1)␈α
to␈α	con␈α␈struct␈α	a␈↓ π%␈ε#[␈↓ π4␈ε#0,␈αε1␈α␈)␈α
sequ␈α␈enc␈α␈e␈α
tha␈α␈t␈α
is␈α
3-distrib␈α␈uted␈α␈,
␈β
J␈↓ βx␈ε&1␈↓ ∧L␈ε&3
␈β
L␈↓ ↓H␈ε#fo␈α␈r␈αw␈α↓h␈α␈ich␈↓ α\␈ε#P␈↓ αr␈ε#r␈↓ α␈␈ε#(␈↓ β
␈ε)U␈↓ βJ␈ε7∃␈↓ ∧
␈ε#)␈α	=␈↓ ∧↑␈ε#.
␈β
X␈↓ β!␈ε&2␈ε,n
␈β
]␈↓ βx␈ε&2␈↓ ∧L␈ε&4
␈β
←␈↓ βx␈∧
←βxα∞␈↓ ∧L␈∧
←∧Lα∞
␈β
␈␈↓ ↓V␈ε329.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈34␈↓ αt␈ε#]␈α⊗Let␈↓ βP␈ε)X␈↓ βz␈ε#,␈↓ ∧∂␈ε)X␈↓ ∧:␈ε#,␈↓ ∧N␈ε#.␈αε.␈αε.␈↓ ¬␈ε#b␈α␈e␈αa␈α(2␈↓ ¬d␈ε)k␈↓ ¬v␈ε#)-distribu␈α␈ted␈αb␈α␈inary␈α
sequ␈α␈en␈α␈ce.␈α∂Sho␈α␈w␈αth␈α␈at
␈β∞
␈↓ βk␈ε&0␈↓ ∧*␈ε&1
␈β∞H␈↓ ε`␈ε↓∩␈↓ πZ␈ε↓∪␈↓ πp␈ε↓≡
␈β∞L␈↓ ε_␈ε#1␈↓ εv␈ε#2␈↓ πε␈ε)k␈↓ π ␈ε7␈␈ε#␈αλ1
␈β∞[␈↓ λ&␈ε&2␈↓ λ4␈ε,k
␈β∞c␈↓ ∧'␈∧∞c∧'α$␈↓ ∧'␈ε#Pr␈↓ ∧J␈ε#(␈↓ ∧U␈ε)X␈↓ ¬~␈ε#=␈α	0)␈ε7␈α	∀␈↓ ε3␈ε#+␈↓ λ∃␈ε#2␈↓ λD␈ε#.
␈β∞n␈↓ ∧q␈ε&2␈ε,␈α␈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␈↓ ∧J␈ε#(␈↓ ∧U␈ε)X␈↓ ¬~␈ε#=␈α
0␈α␈)␈α
=␈↓ ε4␈ε#+␈↓ λ∃␈ε#2␈↓ λD␈ε#.
␈β⊂A␈↓ ∧q␈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↓\␈↓ ε2␈ε∞W␈α↓HA␈α}T␈α
IS␈α	A␈α
RA␈α↓NDOM␈α
S␈α␈E␈α↓QUENCE␈α↓?
␈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␈enc␈α␈es␈αexist␈αtha␈α␈t␈αsa␈α␈tisfy␈αDe|n␈α␈iti␈α↓o␈α␈n␈αR5,␈αy␈α␈et␈↓ 
~␈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>␈α0,␈α
w␈α↓h␈α␈ere␈↓ ∧β␈ε)↔␈↓ ∧4␈ε#is␈αthe␈αn␈α␈u␈α␈m␈α␈ber␈αof␈↓ ε4␈ε)j␈↓ εO␈ε#<␈ε)␈α
n␈ε#␈α
fo␈α␈r␈α
which␈↓ λ5␈ε)U␈↓ λj␈ε#<␈↓ 	,␈ε#.␈α≥(Th␈α␈i␈α↓s␈αmigh␈α}t␈α
be
␈βαX␈↓ ∧∃␈ε,n␈↓ λL␈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␈↓ ∧[␈ε7i␈ε#␈αis␈α
a␈α
\ran␈α␈do␈α␈m"␈ε)␈α
b␈ε#␈α↓-a␈α␈ry␈α
sequ␈α␈en␈α␈ce␈α
acco␈α␈rding␈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␈↓ α␈ε7i␈α↓R␈ε#␈α␈,␈α
sh␈α↓o␈α}w␈α
th␈α␈at␈αthe␈αlatte␈α␈r␈α
su␈α␈bse␈α␈que␈α␈nce␈αis␈αn␈α↓o␈α␈t␈α
o␈α␈nly␈α1-d␈α␈istri␈α↓b␈α␈ut␈α␈ed,␈α
it␈αis␈α
\␈α␈ran␈α␈dom␈α␈"␈αby
␈β∧↓␈↓ ↓n␈ε,n
␈β∧≥␈↓ ↓H␈ε#De␈α␈|n␈α␈i␈α↓tion␈α
R5.
␈β∧P␈↓ ↓V␈ε333.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈22␈↓ αt␈ε#]␈α⊗Let␈ε7␈αh␈↓ β[␈ε)U␈↓ ∧⊂␈ε7i␈ε#␈αan␈α␈d␈ε7␈αh␈↓ ∧r␈ε)U␈↓ ¬(␈ε7i␈ε#␈αb␈α␈e␈αin|␈α␈nite␈αd␈α␈isjoin␈α␈t␈αsu␈α␈bse␈α␈que␈α␈nces␈α
of␈αa␈α
sequ␈α␈enc␈α␈e␈ε7␈αh␈↓ 
o␈ε)U␈↓ _␈ε7i␈ε#.
␈β∧[␈↓ βq␈ε,r␈↓ ¬λ␈ε,s␈↓ ε␈ε,n
␈β∧b␈↓ β⎇␈ε-n␈↓ ¬∃␈ε-n
␈β∧w␈↓ ↓H␈ε#(Th␈α}us,␈↓ α5␈ε)r␈↓ αb␈ε#<␈↓ β∪␈ε)r␈↓ β@␈ε#<␈↓ βq␈ε)r␈↓ ∧≡␈ε#<␈↓ ∧O␈ε7↓␈α¬↓␈αε↓␈↓ ¬∧␈ε#and␈↓ ¬I␈ε)s␈↓ ¬w␈ε#<␈↓ ε(␈ε)s␈↓ εW␈ε#<␈↓ πλ␈ε)s␈↓ π6␈ε#<␈↓ πg␈ε7↓␈αε↓␈α¬↓␈↓ λ≥␈ε#a␈α␈re␈α∂inc␈α␈reasing␈α∞seq␈α␈uen␈α␈ces␈α∂o␈α␈f
␈β¬β␈↓ αC␈ε&0␈↓ β!␈ε&1␈↓ β␈␈ε&2␈↓ ¬Y␈ε&0␈↓ ε8␈ε&1␈↓ π↔␈ε&2
␈β¬∨␈↓ ↓H␈ε#in␈α}tegers␈αan␈α␈d␈↓ β∧␈ε)r␈↓ β6␈ε7≤␈↓ βa␈ε)s␈↓ ∧∂␈ε#for␈αan␈α}y␈ε)␈αm␈ε#␈α␈,␈ε)␈αεn␈ε#.␈α↓)␈α~Let␈ε7␈αh␈↓ ε;␈ε)U␈↓ εo␈ε7i␈ε#␈αbe␈αthe␈αcom␈α}bine␈α␈d␈αsub␈α␈sequ␈α␈enc␈α␈e,␈αso␈αtha␈α␈t
␈β¬*␈↓ β∪␈ε,m␈↓ βq␈ε,n␈↓ εQ␈ε,t
␈β¬1␈↓ ε\␈ε-n
␈β¬F␈↓ ↓H␈ε)t␈↓ ↓p␈ε#<␈↓ α∨␈ε)t␈↓ αH␈ε#<␈↓ αv␈ε)t␈↓ β∨␈ε#<␈↓ βM␈ε7↓␈αε↓␈αε↓␈↓ ∧↓␈ε#an␈α␈d␈α
the␈α
set␈ε7␈α
f␈↓ ¬G␈ε)t␈↓ ¬e␈ε7g␈ε#␈α
=␈ε7␈α
f␈↓ εB␈ε)r␈↓ εc␈ε7g␈αλ[␈α	f␈↓ π.␈ε)s␈↓ πP␈ε7g␈ε#␈α␈.␈α⊗Sho␈α␈w␈α
that␈α
if␈↓ 	G␈ε#P␈↓ 	]␈ε#r␈↓ 	j␈ε#(␈↓ 	u␈ε)U␈↓ 
8␈ε72␈ε)␈α
A␈ε#)␈α
=
␈β¬R␈↓ ↓T␈ε&0␈↓ α+␈ε&1␈↓ βα␈ε&2␈↓ ¬S␈ε,n␈↓ εP␈ε,n␈↓ π>␈ε,n␈↓ 
␈ε,r
␈β¬Y␈↓ 
_␈ε-n
␈β¬n␈↓ ↓H␈ε#P␈↓ ↓↑␈ε#r␈↓ ↓k␈ε#(␈↓ ↓v␈ε)U␈↓ α6␈ε72␈ε)␈α	A␈ε#)␈α	=␈ε)␈α
p␈ε#,␈αthe␈α␈n␈↓ ∧≥␈ε#P␈↓ ∧3␈ε#r␈↓ ∧@␈ε#(␈↓ ∧K␈ε)U␈↓ ¬	␈ε72␈ε)␈α	A␈ε#)␈α
=␈ε)␈α	p␈ε#.
␈β¬z␈↓ α
␈ε,s␈↓ ∧b␈ε,t
␈βε␈↓ α→␈ε-n␈↓ ∧m␈ε-n
␈βε≡␈↓ ↓;␈ε↓x
␈βε ␈↓ ↓S␈ε33␈α␈4.␈↓ α␈ε#[␈ε)M2␈α␈5␈↓ α\␈ε#]␈α⊗De|n␈α␈e␈α
su␈α␈bseq␈α␈uen␈α␈ce␈α
ru␈α␈les␈↓ ¬q␈ε7R␈↓ ε~␈ε#,␈↓ ε1␈ε7R␈↓ εZ␈ε#,␈↓ εq␈ε7R␈↓ π~␈ε#,␈↓ π1␈ε#.␈αε.␈αε.␈↓ πe␈ε#su␈α␈ch␈α
th␈α␈at␈α
Algo␈α␈ri␈α↓t␈α␈hm␈α
W␈α
can␈αbe
␈βε,␈↓ ε␈ε&1␈↓ εK␈ε&2␈↓ π␈ε&3
␈βεH␈↓ ↓H␈ε#u␈α␈sed␈α⊂wi␈α↓th␈α⊂th␈α␈ese␈α⊃ru␈α␈l␈α↓e␈α␈s␈α⊃to␈α⊃giv␈α}e␈α⊃an␈α⊂e{e␈α␈ctiv␈α␈e␈α⊃a␈α␈l␈α↓g␈α␈orithm␈α⊂to␈α⊃co␈α␈nstru␈α␈ct␈α⊃a␈↓ 	S␈ε#[␈↓ 	c␈ε#0␈α␈,␈αε1)␈α⊃se␈α␈que␈α␈nce
␈βεp␈↓ ↓H␈ε#sa␈α␈tisfying␈αDe␈α␈|n␈α␈i␈α↓tion␈α
R1.
␈βπ ␈↓ ↓;␈ε↓x
␈βπ"␈↓ ↓S␈ε33␈α␈5.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈35␈↓ αt␈ε#]␈α⊗(D.␈αW␈α↓.␈↓ ∧α␈ε#L␈α↓o␈α}v␈α␈elan␈α␈d.)␈α≤S␈α␈h␈α↓o␈α}w␈α
th␈α␈at␈αif␈α
a␈αb␈α␈i␈α↓n␈α␈ary␈αse␈α␈qu␈α␈ence␈ε7␈αh␈↓ 	↔␈ε)X␈↓ 	E␈ε7i␈ε#␈αis␈α
R␈α␈5-ran␈α␈do␈α␈m,
␈βπ.␈↓ 	2␈ε,n
␈βπG␈↓ 
\␈ε&1
␈βπJ␈↓ ↓H␈ε#a␈α␈nd␈α	if␈ε7␈α
h␈↓ α0␈ε)s␈↓ αR␈ε7i␈ε#␈α
is␈α
an␈α}y␈α	comp␈α␈uta␈α␈ble␈α	sequ␈α␈enc␈α␈e␈α
as␈α	i␈α↓n␈α	De|␈α␈nition␈α	R4,␈α
th␈α␈en␈↓ λk␈∧πJλkα$␈↓ λk␈ε#Pr␈↓ 	∞␈ε#(␈↓ 	→␈ε)X␈↓ 	↑␈ε#=␈α	1)␈ε7␈α	∃␈↓ 
w␈ε#a␈α␈nd
␈βπU␈↓ α@␈ε,n␈↓ 	5␈ε,s
␈βπZ␈↓ 
\␈ε&2
␈βπ\␈↓ 	B␈ε-n
␈βπ]␈↓ 
\␈∧π]
\α∞
␈βπo␈↓ β8␈ε&1
␈βπq␈↓ ↓H␈ε#Pr␈↓ ↓k␈ε#(␈↓ ↓v␈ε)X␈↓ α:␈ε#=␈α
1␈α␈)␈ε7␈α
∀␈↓ βJ␈ε#.
␈βπ⎇␈↓ α⊃␈ε,s
␈βλα␈↓ β8␈ε&2
␈βλβ␈↓ α≡␈ε-n
␈βλ∧␈↓ β8␈∧λ∧β8α∞
␈βλ⊃␈↓ ↓H␈∧λ⊃↓Hα$
␈βλ$␈↓ ↓V␈ε336.␈↓ α␈ε#[␈ε)H␈α⎇M␈α␈30␈↓ αt␈ε#]␈α⊗Let␈ε7␈αh␈↓ β[␈ε)X␈↓ ∧	␈ε7i␈ε#␈αbe␈αa␈αb␈α␈i␈α↓n␈α␈ary␈α
sequ␈α␈enc␈α␈e␈αth␈α␈at␈αis␈α\ran␈α␈do␈α␈m"␈αac␈α␈cord␈α␈i␈α↓n␈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␈↓ ¬w␈ε7i␈ε#␈αde␈α␈|n␈α␈ed␈αin␈αb␈α␈inar␈α␈y␈αn␈α↓o␈α␈tation␈α
by␈α
the␈αsch␈α␈eme
␈βλW␈↓ ¬e␈ε,n
␈β	↔␈↓ ¬_␈ε)U␈↓ ¬G␈ε#=␈α
(0.␈↓ ε↔␈ε)X␈↓ εB␈ε#)
␈β	"␈↓ ¬/␈ε&0␈↓ ε2␈ε&0␈↓ εM␈ε&2
␈β	M␈↓ ¬_␈ε)U␈↓ ¬G␈ε#=␈α
(0.␈↓ ε↔␈ε)X␈↓ εB␈ε)X␈↓ εl␈ε#)
␈β	X␈↓ ¬/␈ε&1␈↓ ε2␈ε&1␈↓ ε]␈ε&2␈↓ εw␈ε&2
␈β
β␈↓ ¬_␈ε)U␈↓ ¬G␈ε#=␈α
(0.␈↓ ε↔␈ε)X␈↓ εB␈ε)X␈↓ εl␈ε)X␈↓ π↔␈ε#)
␈β
∞␈↓ ¬/␈ε&2␈↓ ε2␈ε&3␈↓ ε]␈ε&4␈↓ πλ␈ε&5␈↓ π"␈ε&2
␈β
9␈↓ ¬_␈ε)U␈↓ ¬G␈ε#=␈α
(0.␈↓ ε↔␈ε)X␈↓ εB␈ε)X␈↓ εl␈ε)X␈↓ π↔␈ε)X␈↓ πB␈ε#)
␈β
D␈↓ ¬/␈ε&3␈↓ ε2␈ε&6␈↓ ε]␈ε&7␈↓ πλ␈ε&8␈↓ π2␈ε&9␈↓ πM␈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|n␈α␈iti␈α↓o␈α␈n␈αR6␈α␈.
␈βc␈↓ ↓V␈ε337.␈↓ α␈ε#[␈ε)M3␈α␈7␈↓ α\␈ε#]␈α⊗(D.␈αCopp␈α␈ersmith␈α␈.␈α↓)␈α~De|␈α␈ne␈αa␈αseq␈α␈uen␈α␈ce␈αth␈α␈at␈αsatis|e␈α␈s␈αDe|n␈α␈ition␈αR4␈αb␈α␈ut␈αn␈α↓o␈α␈t
␈β␈↓ ↓H␈ε#De␈α␈|n␈α␈i␈α↓tion␈απR5.␈α
[␈ε0Hin␈α␈t:␈ε#␈α
Co␈α␈nside␈α␈r␈αλcha␈α␈ngin␈α␈g␈↓ ε␈ε)U␈↓ ε2␈ε#,␈↓ εD␈ε)U␈↓ εi␈ε#,␈↓ ε{␈ε)U␈↓ π!␈ε#,␈↓ π3␈ε)U␈↓ πY␈ε#,␈↓ πk␈ε#.␈αε.␈α¬.␈↓ λ~␈ε#in␈απa␈αλtru␈α␈l␈α↓y␈απran␈α␈do␈α␈m␈αλsequ␈α␈en␈α␈ce.]
␈β⊗␈↓ ε#␈ε&0␈↓ εZ␈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␈α␈.␈↓ ∧∞␈ε#S␈α␈chmid␈α␈t.)␈α"Let␈↓ ¬⎇␈ε)U␈↓ ε6␈ε#b␈α␈e␈α∂a␈↓ πε␈ε#[␈↓ π⊗␈ε#0␈α␈,␈αε1)␈α∂seq␈α␈uen␈α␈ce,␈α⊃a␈α␈nd␈α∂let␈↓ 	q␈ε)↔␈↓ 
∃␈ε#(␈ε)u␈ε#)␈α⊂b␈α␈e␈α∂the
␈β∞→␈↓ ε∀␈ε,n␈↓ 
α␈ε,n
␈β∞5␈↓ ↓H␈ε#n␈α}um␈α␈b␈α␈er␈αof␈α
non␈α␈neg␈α␈ativ␈α␈e␈αi␈α↓n␈α}tege␈α␈rs␈↓ ¬"␈ε)j␈↓ ¬=␈ε7∀␈ε)␈αn␈ε#␈α
such␈αth␈α␈at␈α
0␈ε7␈α∀␈↓ πl␈ε)U␈↓ λ≤␈ε#<␈ε)␈α
u␈ε#.␈α∀Pro␈α␈v␈α}e␈α
th␈α␈at␈αthere␈αis␈α
a
␈β∞A␈↓ λβ␈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␈↓ 	b␈ε7i␈ε#,␈αw␈α␈e␈αh␈α␈a␈α␈v␈α}e
␈β∞i␈↓ 	P␈ε,n
␈β∂,␈↓ ¬~␈ε7j␈↓ ¬#␈ε)↔␈↓ ¬G␈ε#(␈ε)u␈ε#)␈ε7␈αλ␈␈ε)␈απu␈α↓n␈ε7j␈ε#␈α
>␈ε)␈α	c␈↓ π→␈ε#ln␈↓ π:␈ε)N
␈β∂8␈↓ ¬4␈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␈ta␈α␈i␈α↓n␈α
just␈αon␈α␈e␈αmisp␈α␈ri␈α↓n␈α}t?
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