Diophantine approximation and transcendence

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Combinatorics, Automata and Number Theory CANT Edited by Valerie Berthe LIRMM - Universite Montpelier II - CNRS UMR 5506 161 rue Ada, F-34392 Montpellier Cedex 5, France Michel Rigo Universite de Liege, Institut de Mathematiques Grande Traverse 12 (B 37), B-4000 Liege, Belgium

simultaneous diophantine

thue-siegel-roth- schmidt method

real numbers

universite de lyon

infinite string

unique infinite

thue-morse word

diophantine approximation

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Publié par

profil-urra-2012

Nombre de lectures

Langue

English

Combinatorics,Automata

andNumberTheory

TNAC

Editedby
Vale´rieBerthe´
LIRMM-Universite´MontpelierII-CNRSUMR5506
161rueAda,F-34392MontpellierCedex5,France

MichelRigo
GrUannidveerTsritae´vedreseLi1e`2ge(,BIn3s7t)i,tuBt-d4e000MaLtihe`e´gem,atBiqelugeisum

8TranscendenceandDiophantine
approximation
BorisAdamczewski
CNRS,Universite´deLyon,Universite´Lyon1,InstitutCamilleJordan,
43boulevarddu11novembre1918,F-69622Villeurbannecedex,France
YannBugeaud
IRMA-Universite´deStrasbourg-Mathe´matiques-CNRSUMR7501
7rueRene´Descartes,F-67084Strasbourgcedex,France.
Theaimofthischapteristopresentseveralnumber-theoreticproblemsthat
revealafruitfulinterplaybetweencombinatoricsonwordsandDiophantine
approximation.FiniteandinﬁnitewordsoccurnaturallyinDiophantine
approximationwhenweconsidertheexpansionofarealnumberinaninte-
gerbase
b
oritscontinuedfractionexpansion.Conversely,withaninﬁnite
word
a
ontheﬁnitealphabet
{
0
,
1
,...,b
!
1
}
weassociatetherealnumber
!
a
whosebase-
b
expansionisgivenby
a
.Aswell,withaninﬁniteword
a
ontheinﬁnitealphabet
{
1
,
2
,
3
,...
}
,weassociatetherealnumber
"
a
whose
continuedfractionexpansionisgivenby
a
.Itturnsoutthat,iftheword
a
enjoyscertaincombinatorialpropertiesinvolvingrepetitiveorsymmetric
patterns,thenthisgivesinterestinginformationonthearithmeticalnature
andontheDiophantinepropertiesoftherealnumbers
!
a
and
"
a
.
Weillustrateourresultsbyconsideringtherealnumbersassociatedwith
twoclassicalinﬁnitewords,the
Thue-Morseword
andthe
Fibonacciword
,
seeExample1.2.21and1.2.22.Thereareseveralwaystodeﬁnethem.Here,
weemphasizethefactthattheyareﬁxedpointsofmorphisms.
Considerthemorphism
#
deﬁnedonthesetofwordsonthealphabet
{
0
,
1
}
by
#
(0)=01and
#
(1)=0.Then,wehave
#
2
(0)=010
,
#
3
(0)=
01001
,
#
4
(0)=01001010,andthesequence(
#
k
(0))
k
!
0
convergestothe
Fibonacciword
f
=010010100100101001010
∙∙∙
.
(8.1)
Considerthemorphism
$
deﬁnedoverthesamealphabetby
$
(0)=01
and
$
(1)=10.Then,wehave
$
2
(0)=0110
,
$
3
(0)=01101001,andthe
sequence(
$
k
(0))
k
!
0
convergestotheThue-Morseword
t
=011010011001011010010
∙∙∙
.
(8.2)
Forevery
n
"
1,wedenoteby
f
n
the
n
thletterof
f
andby
t
n
the
n
th
424

TranscendenceandDiophantineapproximation
425
letterof
t
.Foraninteger
b
"
2,weset
!
f
=
f
nn
!b1n!dna!
t
=
t
n
.
!nb1n!WefurtherdeﬁneFibonacciandThue-Morsecontinuedfractions,but,since
0cannotbeapartialquotient,wehavetowriteourwordsonanother
alphabetthan
{
0
,
1
}
.Wetaketwodistinctpositiveintegers
a
and
b
,set
f
n
"
=
a
if
f
n
=0and
f
n
"
=
b
otherwise,and
t
"
n
=
a
if
t
n
=0and
t
"
n
=
b
otherwise.Then,wedeﬁne
"
f
!
=[
a,b,a,a,b,a,b,a,...
]=[
f
1
"
,f
2
"
,f
3
"
,f
4
"
,...
]
dna"
t
!
=[
a,b,b,a,b,a,a,b,...
]=[
t
"
1
,t
"
2
,t
"
3
,t
"
4
,...
]
.
Amongotherresults,wewillexplainhowtocombinecombinatorialprop-
ertiesoftheFibonacciandThue-MorsewordswiththeThue-Siegel-Roth-
Schmidtmethodtoprovethatallthesenumbersaretranscendental.Be-
yondtranscendence,wewillshowthattheFibonaccicontinuedfractions
satisfyaspectacularpropertiesregardingaclassicalprobleminDiophan-
tineapproximation:theexistenceofrealnumbers
!
withthepropertythat
!
and
!
2
areuniformlysimultaneouslyverywellapproximablebyrational
numbersofthesamedenominator.Wewillalsodescribean
adhoc
con-
structiontoobtainexplicitexamplesofpairsofrealnumbersthatsatisfy
theLittlewoodconjecture,whichisamajoropenprobleminsimultaneous
Diophantineapproximation.
WeusethefollowingconventionthroughoutthisChapter.TheGreek
letter
!
standsforarealnumbergivenbyitsbase-
b
expansion,where
b
alwaysmeansanintegeratleastequalto2.TheGreekletter
"
stands
forarealnumbergivenbyitscontinuedfractionexpansion.Ifitspartial
quotientstakeonlytwodi
!
erentvalues,thesearedenotedby
a
and
b
,which
representdistinctpositiveintegers(here,
b
isnotassumedtobeatleast2).
Theresultspresentedinthischapterarenotthemostgeneralstatements
thatcanbeestablishedbythemethodsdescribedhere.Ourgoalisnotto
makeanexhaustivereviewofthestate-of-the-art,butrathertoemphasize
theideasused.Theinterestedreaderisdirectedtotheoriginalpapers.

426
B.Adamczewski,Y.Bugeaud
8.1Theexpansionofalgebraicnumbersinanintegerbase
Throughoutthepresentsection,
b
alwaysdenotesanintegeratleastequal
to2and
!
isarealnumberwith0
<
!
<
1.Recallthatthereexistsa
uniqueinﬁniteword
a
=
a
1
a
2
∙∙∙
deﬁnedovertheﬁniteset
{
0
,
1
,...,b
!
1
}
,
calledthe
base-
b
expansion
of
!
,suchthat
an!
=
b
n
:=0
.a
1
a
2
∙∙∙
,
(8.3)
!1n!withtheadditionalconditionthat
a
doesnotterminateinaninﬁnitestring
ofthedigit
b
!
1.Obviously,
a
dependson
!
and
b
,butwechoosenotto
indicatethisdependence.
Forinstance,inbase10,wehave
3
/
7=0
.
(428571)
!
dna%
!
3=0
.
314159265358979323846264338327
∙∙∙
.
Conversely,if
a
=
a
1
a
2
∙∙∙
isaninﬁniteworddeﬁnedovertheﬁnite
alphabet
{
0
,
1
,...,b
!
1
}
suchthat
a
doesnotterminateinaninﬁnite
stringofthedigit
b
!
1,thereexistsauniquerealnumber,denotedby
!
a
,
suchthat
!
a
:=0
.a
1
a
2
∙∙∙
.
Thisnotationdoesnotindicateinwhichbase
!
a
iswritten.However,this
willbeclearfromthecontextandshouldnotcauseanydi
"
culty.
Inthesequel,wewillalsosometimesmakeaslightabuseofnotationand,
givenaninﬁniteword
a
deﬁnedovertheﬁnitealphabet
{
0
,
1
,...,b
!
1
}
that
couldendinaninﬁnitestringof
b
!
1,wewillwrite
0
.a
1
a
2
∙∙∙
todenotetheinﬁnitesum
an.!nb1n!Werecallthefollowingfundamentalresultthatcanbefoundintheclas-
sicaltextbook(HardyandWright1985).
Theorem8.1.1
Arealnumberisrationalif,andonlyif,itsbase-
b
expan-
sioniseventuallyperiodic.

TranscendenceandDiophantineapproximation
427
8.1.1Normalnumbersandalgebraicnumbers
Atthebeginningofthe20thcentury,E´mileBorel(Borel1909)investigated
thefollowingquestion:
Howdoesthedecimalexpansionofarandomlychosenrealnumberlook
?ekilThisquestionleadstothenotionofnormality.
Deﬁnition8.1.2
Arealnumber
!
iscalled
normaltobase
b
if,forany
positiveinteger
n
,eachoneofthe
b
n
wordsoflength
n
onthealphabet
{
0
,
1
,...,b
!
1
}
occursinthebase-
b
expansionof
!
withthesamefrequency
1
/b
n
.Arealnumberiscalleda
normalnumber
ifitisnormaltoevery
integerbase.
E´.Borel(Borel1909)provedthefollowingfundamentalresultregarding
normalnumbers.
Theorem8.1.3
ThesetofnormalnumbershasfullLebesguemeasure.
Someexplicitexamplesofrealnumbersthatarenormaltoagivenbase
areknownforalongtime.Forinstance,thenumber
0
.
123456789101112131415
∙∙∙
,
(8.4)
whosesequenceofdigitsistheconcatenationofthesequenceofallpositive
integerswritteninbase10andrangedinincreasingorder,wasprovedtobe
normaltobase10in1933byD.G.Champernowne(Champernowne1933).
Incontrast,todecidewhetheraspeciﬁcnumber,like
e
,
%
or
#2=1
.
414213562373095048801688724209
∙∙∙
,
isorisnotanormalnumberremainsachallengingopenproblem.Inthis
direction,thefollowingconjectureiswidelybelievedtobetrue.
Conjecture8.1.4
Everyrealirrationalalgebraicnumberisanormalnum-
.reb

8.1.2Complexityofrealnumbers
Conjecture8.1.4isreputedtobeoutofreach.Wewillthusfocusour
attent

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