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β -expansions for cubic Pisot numbers
Frédérique Bassino
To cite this version:
Frédérique Bassino. β-expansions for cubic Pisot numbers. 5th Latin American Theoretical INfor-
matics (LATIN’2002), 2002, United States. pp.141-152. �hal-00619858�
FrederiqueBassino
I.G.M.,UniversitedeMarneLa Vallee
77454 Marne-la-ValleeCedex2.Frane
e-mail:bassinouniv-mlv.fr
Abstrat. Realnumbersanberepresentedinanarbitrarybase>1
usingthetransformation T:x!x (mod1) oftheunitinterval;any
real number x 2 [0;1℄ is then expanded into d
(x) = (x
i )
i1 where
xi=bT i 1
(x).
The losure of the set of the expansions of real numbers of [0;1[ is a
subshift of fa 2 N j a < g N
, alled the beta-shift. This dynamial
system is haraterizedby the beta-expansionof 1; inpartiular, it is
of nite type if and only if d
(1) is nite; is then alled a simple
beta-number.
Werst omputethe beta-expansionof 1for any ubiPisot number.
Nextweshowthatubisimplebeta-numbersarePisotnumbers.
Introdution
Representationsofreal numbersin anarbitrarybase >1,alled beta-expan-
sions, havebeenintrodued by Renyi ([14℄). Theyarise from the orbitsof the
pieewise-monotonetransformationof theunitinterval:T
:x!x (mod1).
Suhtransformationswere extensivelystudiedin ergoditheory([13℄).
Morepreisely,anyrealnumberx 2[0;1℄isexpandedinto d
(x)=(x
i )
i1
where x
i
=bT i 1
(x). The nonnegativeintegersd
i
are elementsof the digit
alphabet A = fa 2 N j a < g. These representations generalize standard
representationsin anintegralbaseto arealbase;indeedthe beta-expansionof
anyreal numberof[0;1[an equivalentlybeobtainedbythegreedyalgorithm.
Onlythebeta-expansionof1diers.
Propertiesofbeta-expansionsarestronglyrelatedtosymbolidynamis([4℄).
The losure of the set of innite sequenes, appearing as beta-expansions of
numbers of the interval [0;1[, is a dynamial system, that is, a losed shift-
invariantsubsetofA N
,alled thebeta-shift.
An importantproperty ofthebeta-shiftis that itsnature is entirelydeter-
mined,in aombinatorialmanner,bythebeta-expansionof1:thebeta-shiftis
so,thatistosaythesetofitsnitefatorsisreognizedbyaniteautomaton,
ifandonlythebeta-expansionof1iseventuallyperiodi([3℄);itisofnitetype,
that is tosay theset ofitsnite fatorsis dened byforbiddinganite set of
words,ifandonlyifthebeta-expansionof1isnite([12℄).
Whenthebeta-expansionof1iseventuallyperiodi,isalledabeta-number
trand([3℄)andbyShmidt([15℄).Inpartiular,itisknownthatPisotnumbers
arebeta-numbers.ConerningSalemnumbers,weonlyknowthatifisaSalem
numberofdegree4,thenthebeta-expansionof1iseventuallyperiodi([5℄). It
is onjetured that Salem numbersof degree6are still beta-numbers,but not
allSalemnumbersofdegree8([7℄).
ThedomainoftheGaloisonjugatesofallbeta-numberswasalsoinvestigated
independentlybySolomyak([16℄)andbyFlatto,LagariasandPoonen([8℄).
Forageneralpresentationofthebeta-shiftoneanreferto[9℄.
Inthefollowing,wesummarizepropertiesofbeta-numbers.Weomputethe
beta-expansionof1foranyubiPisotnumberandweestablishaharateriza-
tionofubisimplebeta-numbers,showingthattheyarePisotnumbers.
Averyloseproblem,seenfromthepointofviewofnumerationsystems,was
studiedbyAkiyama([1℄).Heshowedthatintheubiase,therealnumbersof
theset N[
1
℄haveanite beta-expansionif andonly is aPisotunit and1
has anitebeta-expansion.This niteness problemis equivalent to aproblem
offrataltilinggeneratedbyPisotnumbers.
1 Beta-numbers
Real numbers an be representedin an arbitrary base > 1using the trans-
formationT
:x !x (mod1) oftheunit interval;anyrealnumberx2[0;1℄
is then expanded into d
(x) =(x
i )
i1
where x
i
= bT i 1
(x). When abeta-
expansionisoftheformuv
!
, theexpansionissaidto beeventuallyperiodi. If
arepresentationends withinnitelymanyzeros,likeu0
!
,itissaidtobenite
andtheendingzerosareomitted.
Let us denote by S
the losure of all beta-expansions of real numbers of
[0;1[and by the(one-sided) shiftdened by((x
i )
i1 )=(x
i+1 )
i1
.The set
S
endowedwiththeshiftisalledthebeta-shift,itisasubshiftofA N
,Abeing
thedigitset,i.e., A=fa2N ja<g.
Animportantproperty([13℄)ofthebeta-shiftS
isthatitsnatureisentirely
determined by d
(1) the beta-expansion of 1.Indeed, setting d
(1) =d
(1) if
d
(1)isinniteandd
(1)=(d
1 d
2 :::d
n 1 (d
n 1))
!
ifd
(1)=d
1 d
2 :::d
n 1 d
n ,
asequenexofnonnegativeintegersbelongstoS
ifandonlyifitsatisesthe
followinglexiographialorderonditions:8p0;
p
(x)d
(1).
Reallthatthebeta-expansionof1alsoanbeharaterized([13℄)bylexio-
graphialorderonditions:letd=(d
i )
i1
beasequeneofnonnegativeintegers
dierentfrom10
!
,suhthat P
i1 d
i
i
=1,withd
1
1andfori2,d
i d
1 ,
thendisthebeta-expansionof1ifandonlyifforallp1, p
(d)<d.
Wereallthatanalgebraiinteger stritlygreaterthan1isalledaPerron
number if all its Galois onjugates have modulus stritly less than , aPisot
number if all its Galois onjugates have modulus stritly less than 1, and a
Salem number ifall itsonjugatesare lessthan 1in modulus and at leastone
Let beabeta-number.Denotebyd
(1)=d
1 :::d
n (d
n+1 :::d
n+p
) ,where
nandparehosenminimal,thebeta-expansionof1.Thentheadjaenymatrix
M
ofthenite automatonreognizing theset ofits nitefators (Fig.1)isa
primitive (i.e., itsassoiatedgraphisstronglyonnetedand thelengthsofits
ylesarerelativelyprime)nonnegativeintegralmatrixwhosespetralradiusis
; so,fromthePerron-Frobeniustheorem, isaPerronnumber.
1 2 3 n n+1 n+p
0;1;:::;d
1 1
d
1
d
2
0;:::;d
n 1 0;:::;d
3 1 0;:::;d
2 1
0;:::;d
n+p 1 0;:::;d
n+1 1
0;:::;d
n+p 1 1
d
n+p 1 d
n+p
Fig.1. AutomatonreognizingthesetofthenitefatorsofS
TheharateristipolynomialofM
P(X)=X n+p
n+p
X
i=1 d
i X
n+p i
X n
+ n
X
i=1 d
i X
n i
isalled,followingtheterminologyintroduedbyHollander([11℄),theassoiated
beta-polynomial.
AsP isamultipleoftheminimalpolynomialM
of,P(0)=d
n+p d
n isa
multipleofjM
(0)j=j
Q
i
j,where
i
runs overthesetofalgebraionjugates
of. So,wegetthat j Q
i
j hasto besmallerthanb.
Asaonsequene,inthequadratiase,theonlybeta-numbersarethePisot
numbers.Conversely,it isknownthat if is aPisotnumberthen isabeta-
number([2℄). An importantgapremainsbetweenPisotandPerronnumbers.
Example 1. Thequadratinumber=(1+ p
13)=2isnotabeta-numbersine
M (X)=X 2
X 3andM (0)>b.
LetbethePisotnumber(3+ 5)=2,thenisabeta-numberandd
=21
!
.
Let bethegolden ratio(1+ p
5)=2, then is asimplebeta-numberand
d
(1)=11.
Ontheotherhand,thedomainoftheGaloisonjugatesofbeta-numberswas
studied bySolomyak ([16℄)and independentlybyFlatto,LagariasandPoonen
([8℄). They showed in partiular that if the beta-expansion of 1 is eventually
periodithentheGaloisonjugatesof havemoduluslessthanthegoldenratio
(1+ p
5)=2.Itwasalreadyknown(see[9℄)thatannothaveaGaloisonjugate
greaterthan1.
Solomyak ([16℄) proved that the topologial losure of onjugates of beta-
numbersandtheoneofonjugatesofsimplebeta-numbersarethesame.How-
ever, there is animportant dierenebetweenthe onjugatesof beta-numbers
and theones of simplebeta numbers:if is asimplebeta-numberthen has
noalgebraionjugatethatisanonnegativerealnumber.
Indeed,let bea simple beta-number and set d
(1) =d
1 :::d
n
. Consider
1 2 3 n−1 n
0;1;:::;d
1 1
0;:::;d
3 1 0;:::;d
2 1
d
n 1
0;:::;d
n 1 1
0;:::;d
n 1 d
1
d
2
Fig.2. AutomatonreognizingthesetofthenitefatorsofS
the nite automaton reognizing the set of the nite fators of the assoiated
beta-shift (Fig. 2). Let M
be the transition matrix of this automaton. The
harateristipolynomialofM
,whihisalledtheassoiatedbeta-polynomial,
P(X)=X n
n
X
i=1 d
i X
n i
Salem number, 1= > 0 is aGalois onjugate of , and so is not a simple
beta-number.
ThepreviousonditionsaresuÆientforaquadratialgebraiintegerto be
asimplebeta-number.
Proposition1. [10℄ The simple beta-numbers of degree 2areexatly the qua-
drati Pisotnumberswithoutapositive real Galois onjugate. Theyarethe pos-
itiverootsofthe polynomials
X 2
aX b with ab1;
The beta-expansionof 1isthend
(1)=ab.
Example 3. The minimalpolynomialof (1+ p
5)=2 isX 2
X 1,(1+ p
5)=2
isasimplebeta-numberandd
(1)=11.
Theminimalpolynomialof(3+ p
5)=2isX 2
3X+1,therefore(3+ p
5)=2
isnotasimplebeta-number.
2 Beta-expansions of 1 for ubi Pisot numbers
LetusrealltheharaterizationofubiPisotnumbersduetoAkiyama([1℄)
Theorem1 (Akiyama [1℄).Let >1be aubinumberandlet
M
(x)=X 3
aX 2
bX
be itsminimalpolynomial.
Then isaPisot numberif andonlyif itbothinequalities
jb 1j<a+ and ( 2
b)<sgn()(1+a)
hold.
Remark 1. Notethat amustbeanonnegativeinteger.
Thefollowingtheoremgivesthe-expansionof1foranyubiPisotnumber.
Theorem2. Let beaubiPisot numberandlet
M
(x)=X 3
aX 2
bX
be itsminimalpolynomial. Then the beta-expansionof1is
{ Case1:Whenba,then d
(1)=(a+1)(b 1 a)(a+ b)(b ).
{ Case2: When0ba, if>0, d
(1)=ab,otherwise,
d (1)=a[(b 1)(+a)℄
!
:
otherwised
(1)=(a 1)(a+b 1)(a+b+ 1)
!
{ Case 4: When b a, let k be the integer of f2;3;:::;a 2g suh that,
denotinge
k
=1 a+(a 2)=k,e
k
b+<e
k 1 .
If b(k 1)+(k 2)(k 2) (k 1)a,d
(1)=d
1 :::d
2k +2 with
d
1
=a 2;
d
k +2 i
= (k+3 i)+a(k+2 i)+b(k+1 i)+(k i);3ik
d
k
= k+ak+b(k 1)+(k 2)
d
k +1
= (k 1)+ak+bk+(k 1)
d
k +2
= (k 2)+a(k 1)+bk+k
d
2k +2 i
= (i 2)+a(i 1)+bi+(i+1) k3;2i(k 1)
d
2k +1
=b+2 and d
2k +2
=:
If b(k 1)+(k 2)>(k 2) (k 1)a,let mbe the integer dened
by m=minfi2N suhthat (i+1)b+i>i (i+1)ag.
Whenm=1,d
(1)=(a 2)(2a+b 2)(2a+2b+ 2)(2a+2b+2 2)
!
.
Whenm>1, d
(1)=d
1 d
2 :::d
m+2 d
!
m+3 ,with
d
1
=a 2; d
2
=2a+b 3;
d
m+3 i
=2a+b 3+(m+1 i)(a+b+ 1) m3;3im;
d
m+1
=2a+b 2+(m 1)(a+b+ 1);
d
m+2
=a+b 1+m(a+b+ 1);
d
m+3
=(m+1)(a+b+ 1):
Example 4. Whenab0and>0,weobtaintheonlybeta-expansionof1
oflength3.
ThesmallestPisotnumberhasM
=X 3
X 1asminimalpolynomial,it
isasimplebeta-numberandd
(1)=10001.
Thepositiveroot of M
=X 3
3X 2
+2X 2isa simplebeta-number
andd
(1)=2102.
Theasewhereb ashowsthatfromaubisimplebeta-number,wean
obtainanarbitrarylongbeta-expansionof 1.Foranyintegerk greaterthan or
equalto2,therealrootoftheirreduiblepolynomialX 3
(k+2)X 2
+2kX k,
isasimplebetanumberwhoseintegerpartisequaltok,andthebeta-expansion
of 1 has length 2k+2. Fork =2, we get d
(1) =221002; for k = 3, we get
d
(1)=31310203.
Example 5. Thegreatestpositiveroot ofM
=X 3
2X 2
X+1isabeta-
numberandd
(1)=2(01)
!
.
If isthepositiverootofX 3
5X 2
+3X 2,thend
(1)=413
!
.When
isthegreatestpositiverootofX 3
5X 2
+X+2,thend
(1)=431
!
.
Foranyintegerkgreaterthanorequalto3,therealroot oftheirreduible
polynomialX 3
(k+2)X 2
+(2k 1)X (k 1),isabetanumberwhoseinteger
1,thelengthofitspreperiodk.Fork=3,wegetd
(1)=3302 ; fork=4,we
getd
(1)=42403
!
.
Proof. It is known that Pisot numbers are beta-numbers, thus, for any ubi
Pisotnumber,thebeta-expansionof 1is niteoreventuallyperiodi. Inany
ase,werstomputetheassoiatedbeta-polynomialP.Nextweprovethatthe
sequened=(d
i )
i1
ofnonnegativeintegersobtainedfromthebeta-polynomial
satisfylexiographialorder onditions:forallp1, p
(d)<d.
Firstofall,wereallthat,fromTheorem1,aubinumber,greaterthan
1andhaving
M
(X)=X 3
aX 2
bX
asminimal polynomial,isaubiPisotnumberifandonlyifitboth
jb 1j<a+ and ( 2
b)<sgn()(1+a)
hold.
DenotebyQtheomplementaryfatorofthebeta-polynomialP denedby
P(X)=M
(X)Q(X).As weshall seein what follows,the valueof Qdepends
uponthevalueoftheoeÆientsofM
.
Case1:Whenb>a,as isaPisotnumber,fromTheorem1,isapositive
integer. In this ase, the omplementary fator is Q(X) = X 2
X +1 and
d
(1)=(a+1)(b 1 a)(a+ b)(b ).
Indeed, as ( 2
b) < sgn()(1+a) and > 0, we get a+1. As
jb 1j<a+,wegetb 1 aaand0a b+.Fromb>a,wegetthat
0b a 1and,asa+1,that a b+a.Finallyas0a b+a,
weobtain0b a.
Case 2:When0ba, theomplementaryfatoristhen Q(X)=1and
theassoiatedbeta-polynomialisequalto theminimalpolynomial.
If>0,thend
(1)=ab.Indeed,as( 2
b)<sgn()(1+a),wegeta.
If < 0, then d
(1) = a[(b 1)(a+)℄
!
. As jb 1j < a+ , we get
b 1a 2.As( 2
b)<sgn()(1+a),wegetthat aand,onsequently,
0+aa 1.
Case 3:When a< b<0, if b+ 0then the omplementary fator is
Q(X) = X +1 and d
(1) = (a 1)(a+b)(b+). Indeed, as a < b < 0,
we obtain 1 a+b a 1. Sine b+ 0, is a positive integer. From
( 2
b)<sgn()(1+a),wegetthata 1and b+a 2.
Ifb+<0,thenQ(X)=1and d
(1)=(a 1)(a+b 1)(a+b+ 1)
!
.
As a<b<0,weget0a+b 1a 2.Fromjb 1j<a+,wegetthat
1a+b+ 1andasb+<0,weobtaina+b+ 1a 2.
Case4:Firstofall,sinejb 1j<a+,weget a+2b+.Moreoveras
b a,weget2andas( 2
b)<sgn()(1+a),weobtaina 2,thus
b+ 2.So,thereexistsanintegerkinf2;3;:::;a 2g,suhthat,denoting
e
k
=1 a+(a 2)=k,e
k
b+<e
k 1 .
Whenb(k 1)+(k 2)(k 2) (k 1)a,theomplementaryfatoris
Q(X)= (X
k
1)(X k +1
1)
2
1 2k +2
d
1
=a 2;
d
k +2 i
= (k+3 i)+a(k+2 i)+b(k+1 i)+(k i);k3;3ik
d
k
= k+ak+b(k 1)+(k 2)
d
k +1
= (k 1)+ak+bk+(k 1)
d
k +2
= (k 2)+a(k 1)+bk+k
d
2k +2 i
= (i 2)+a(i 1)+bi+(i+1) k3;2i(k 1)
d
2k +1
=b+2 and d
2k +2
=:
Wenowverifythatthelexiographialorderonditionsond
(1)aresatised.
As2a 2andb+ 2,wegetd
2k +1
a 4.Frome
k
b+and
b(k 1)+(k 2)(k 2) (k 1)a,wegetd
2k +1 0.
For k3and 2ik 1,d
2k +2 i
= (i 2)+a(i 1)+bi+(i+1).
As b+ <e
i
, we getd
2k +2 i
<. As a+2b+ andb+2 0,weget
d
2k +2 i i.
Ase
k
b+,weobtaind
k +2
0.Sinea 2,d
k +1
>d
k +2
andsine
b+ 2,d
k
>d
k +1
.Moreoverfromb(k 1)+(k 2)(k 2) (k 1)a,
wegetd
k
a 2.
Fork3,asjb 1j<a+,weobtaind
2
<<d
k 1
.Asb+<e
k 1 and
b+20,wegetd
k 1
<a 2.Moreoverfroma 2anda+b+ 1>0,
wegetthat d
2
=2a+b 3isnonnegative.
Alld
i
'saresmallerthand
1 ,onlyd
2k +2 andd
k
anbeequaltod
1
.Therefore
wehavetoverifythatd
2 d
k +1
whenk3(otherwised
2
=d
k andd
k
>d
k +1 ).
Ifd
k
=a 2, thenb+=e
k , andd
k +1
=a 1.Asa+b+ 1>0,we
obtain d
k +1 d
2
. In ase of equality, if k = 3, then d
3
= d
k and d
k
> d
k +2 ,
otherwised
3
>d
2 andd
k +1
>d
k +2
,therefored
3
>d
k +2 .
Solexiographialorderonditionsaresatisedandd
1 :::d
2k +2
isthebeta-
expansionof1.
Whenb(k 1)+(k 2)>(k 2) (k 1)a,asb a,wegetk3.Letm
betheintegerdenedbym=min fi2N suhthat (i+1)b+i>i (i+1)ag.
Notethat bydenitionofm, mk 2andsineb a,m1.Inthisase,
theomplementaryfatoris
Q(X)= m
X
i=0 X
i
:
Thebeta-expansionof1istheneventuallyperiodiwithperiod1,thelength
ofthepreperiodism+2.
Whenm=1,P(X)=X 4
(a 1)X 3
(a+b)X 2
(b+)X and
d
(1)=(a 2)(2a+b 2)(2a+2b+ 2)(2a+2b+2 2)
!
:
Hered
3
=d
m+2
=a+b 1+m(a+b+ 1)andd
4
=d
m+3
=(m+1)(a+b+ 1).
Whenm>1,
P(X)=X m+3
(a 1)X m+2
(a+b 1)X m+1
P
m
i=3
(a+b+ 1)X i
2
andd
(1)=d
1 d
2 :::d
m+2 d
m+3 ,with
d
1
=a 2; d
2
=2a+b 3;
d
m+3 i
=2a+b 3+(m+1 i)(a+b+ 1) m3;3im;
d
m+1
=2a+b 2+(m 1)(a+b+ 1);
d
m+2
=a+b 1+m(a+b+ 1);
d
m+3
=(m+1)(a+b+ 1):
Inbothases,d
1
=a 2.Sineb(k 1)+(k 2)>(k 2) (k 1)aand
a 2,weget 2a+3b.Moreoverasb a,1d
2
a 2whenm=1,
and0d
2
a 3otherwise.Bydenitionofm,(m+1)b+m>m (m+1)a,
thusd
m+2
0andd
m+3
.Sinee
k
b+<e
k 1
andmk 2,weobtain
d
m+3
a 3andd
m+2
a 3.
Whenm>1,sinemb+(m 1)(m 1) ma,wegetd
m+1
a 2.As
02a+b 2anda+b+ 1>0,d
m+1
>0.Moreoverasa+b+ 1>0,
onehasd
2
<d
3
<:::<d
m+1
.Notethat,when m3,d
2
6=a 2.
Wenowstudytheaseswhered
i
isnotstritlysmallerthand
1
.Whenm=1,
onlyd
2
maybeequaltoa 2,thenb= aandd
3
= 2,thusd
3
<d
2 .When
m>1,onlyd
m+1
maybeequaltoa 2,thenmb= ma (m 1)+(m 1),
andthusd
2 d
m+2
=a 1 isapositiveinteger.
Wehaveprovedthat thelexiographialorderonditionsond
(1):
d
1 d
2 :::d
!
m+3
>
lex d
i d
i+1 :::d
!
m+3
for2im+3;
are satised,showingin this waythat theannouned beta-expansionsof 1are
right.
Remark 2. ThepolynomialsQthatappearintheubiaseareylotomi.In
thegeneralase,Qanbenonylotomiandevennonreiproal([6℄).
3 Cubi simple beta-numbers
Inthefollowing,weestablishthatubisimplebeta-numbersarePisotnumbers.
NextwegiveneessaryandsuÆientonditionsontheoeÆientsoftheminimal
polynomialof for tobeasimplebeta-number.
Theorem3. If isaubi simplebeta-number then isa Pisotnumber.
Remark 3. This is nolonger trueforsimplebeta-numbersofdegree 4.Forex-
ample,thepositiverootofX 4
3X 3
2X 2
3isasimplebeta-number,butis
notaPisotnumber.
Proof. Let beaubisimplebeta-numberandlet
M
(X)=X 3
aX 2
bX
beitsminimalpolynomial.Then hasnopositiverealalgebraionjugateand
jjb,diretlyimplies,whentheGaloisonjugatesofarenotrealnumbers,
that isaPisotnumber.
TheonlyotheraseistheasewherebothGaloisonjugates
1 and
2 of
arenegativerealnumbers.Wethenassumethatisaubisimplebeta-number
that is notaPisotnumber,and showthat thesehypothesesare ontraditory.
Let
1 and
2
betheGalois onjugatesof . As0<b,ifoneofthe
i 's
issmallerthan 1theother oneis greaterthan 1.Moreover,asthemodulus
ofaGaloisonjugateofabeta-numberissmallerthanthegoldenratio,onean
suppose,forexample,that
1+ p
5
2
<
2
< 1<
1
<0<
Consequently, M
( 1) > 0,in other words, b > a++1. Note that here
a2fb 2;b 1g.
Asisasimplebeta-number,d
(1)=d
1 d
2 :::d
n
.DenotebyPtheassoiated
-polynomial:
P(X)=X n
n
X
i=1 d
i X
n i
anddenote byQ= P
i0 q
i X
i
thequotientofthedivision upontheinreasing
powersofP byM
.Inotherwords,
P(X)=M
(X)Q(X)
We shall show, by indution,that q
0
1, and that for all i 0, jq
i+1 j> jq
i j
withsgn(q
i+1
)= sgn(q
i
).Weshallonludefromthegrowthofthemoduliof
itsoeÆientsthatQisaninniteseries,andthusthat d
(1)isnotnite.
Inwhatfollows,wemainlyusethefatthatthed
i
'sarenonnegativeintegers
smallerthanbandtheinequalityba++2.
Firstof all, as d
n
= q
0
and d
n
and are positive integers,q
0
1. Sine
d
n 1
=q
0 b+q
1 andq
0 1,d
n 1 q
0 a+2q
0 +(q
0 +q
1
).Whena=b 1,
we diretly get from d
n 1
b, that q
1
< q
0
. When a = b 2, the
lexiographialorderonditionsond
(1)implythat
d
n 1 d
n
<d
1 d
2 :::d
n :
Bydenitionofbeta-expansions,d
1
=bandhered
2
<d
n
.Indeedas
2
= 1
2
a + r
(a ) 2
4
;
and
2
> (1+ p
5)=2,wegetthat
>
p
5 1
+ 1+
p
5
fg;
n
d
2
=bfgisstritly smallerthand
n
. Thereforethepreviouslexiographial
orderonditionimpliesthatd
n 1
<b.So,asd
n 1
b+(q
0 +q
1 ),q
1
< q
0 .
Asd
n 2
=q
0 a + q
1 b + q
2 andq
1
< q
0
<0,d
n 2 (q
1 + q
0 )a + 2q
1 + (q
1 + q
2 ),
that isd
n 2
< b+(q
1 +q
2 ),so q
2
> q
1 .
Forallpositiveintegersi,d
n (2i+1)
= q
2i 2 +q
2i 1 a+q
2i b+q
2i+1
.From
q
2i
>0,wegetd
n (2i+1) (q
2i 1 +q
2i )a+q
2i +(q
2i q
2i 2 )+(q
2i +q
2i+1 ).
From(q
2i 1 +q
2i
)1, q
2i
>2iand (q
2i q
2i 2
)>1, we obtaind
n (2i+1)
>
b+(q
2i +q
2i+1
), andthus q
2i+1
< q
2i .
Forallpositiveintegersi,d
n (2i+2)
= q
2i 1 +q
2i a+q
2i+1 b+q
2i+2
.From
q
2i+1
<0,wegetd
n (2i+1) (q
2i +q
2i+1 )a+q
2i+1 +(q
2i+1 q
2i 1 )+(q
2i+1 +
q
2i+2
).As (q
2i +q
2i+1
) 1,q
2i+1
< (2i+1)and (q
2i+1 q
2i 1
)< 1,we
getd
n (2i+2)
< b+(q
2i+1 +q
2i+2
),thusq
2i+2
> q
2i+1 .
SoQ isan inniteseries; onsequentlyif isnot aPisotnumber,d
(1) is
notnite.
AsaonsequeneofTheorems2and3,weobtaintheaboveharaterization
ofubisimplebeta-numbers.
Proposition2. Let be aubiPisot numberandlet
M
(x)=X 3
aX 2
bX
beits minimalpolynomial.
Then is asimple beta-number if and only itsatises oneof the following
onditions:
{ Case1: b0and>0
{ Case2: a<b<0andb+0
{ Case 3: b a and b(k 1)+(k 2) (k 2) (k 1)a, where k is
the integer in f2;3;:::;a 2g suh that, denoting e
k
=1 a+(a 2)=k,
e
k
b+<e
k 1 .
Theproblemofndingsuhaharaterizationremainsopenforsimplebeta-
numbersof higherdegree.
Referenes
[1℄ S.Akiyama. CubiPisotunitswithnitebetaexpansions. InF.Halter-Kohand
R.F.Tihy,editors,AlgebraiNumberTheory andDiophantine Analysis,11{26.de
Gruyter,2000.
[2℄ A. Bertrand. Developpementsen base de Pisot et repartition modulo 1. C. R.
Aad. Si.Paris,285:419{421,1977.
[3℄ A. Bertrand-Mathis. Developpementen base, repartition modulo1 delasuite
(x n
)
n0
,langagesodeset-shift. Bull.So.Math. Frane,114:271{323,1986.
[4℄ F.Blanhard. -expansionsandsymbolidynamis. Theor.Comput.Si.,65:131{
theory, pages57{64.deGruyter,1989.
[6℄ D.W.Boyd.OnbetaexpansionsforPisotnumbers.MathematisofComputation,
65(214):841{860, 1996.
[7℄ D.W.Boyd. OnthebetaexpansionforSalemnumbersofdegree6. Mathematis
of Computation,65(214):861{875, 1996.
[8℄ L.Flatto,J.Lagarias,andB.Poonen.Thezetafuntionofthebetatransformation.
ErgodiTheoryDynamialSystems,14:237{266, 1994.
[9℄ C. Frougny. Numeration Systems, hapter 7, in M. Lothaire, Algebrai Com-
binatoris on Words. Cambridge University Press, to appear, available at
http://www-igm.univ-mlv.fr/be rste l/Lot hair e/.
[10℄ C. Frougny and B. Solomyak. Finite-expansions. Ergodi Theory Dynamial
Systems, 12:713{723,1992.
[11℄ M. Hollander. Greedynumerationsystemsandregularity. Theory ofComputing
Systems, 31:111{133,1998.
[12℄ S.Ito and Y.Takahashi. Markov subshifts and realization of -expansions. J.
Math. So.Japan,26:33{55, 1974.
[13℄ W.Parry. Onthebetaexpansionsofrealnumbers.AtaMath.Aad.Si.Hung.,
11:401{416, 1960.
[14℄ A. Renyi. Representations for real numbersand their ergodi properties. Ata
Math. Aad.Si.Hung.,8:477{493, 1957.
[15℄ K.Shmidt. OnperiodiexpansionsofPisotnumbersandSalemnumbers. Bull.
LondonMath.So.,12:269{278, 1980.
[16℄ B.Solomyak.Conjugatesofbeta-numbersandthezero-freedomainforalassof
analytifuntions. Pro.LondonMath.So.,68(3):477{498,1994.
