Numeration Systems

CHAPTER

Numeration systems

Contents

Numeration systems

Intro duction

Standard representation of numb ers

Representation of integers

Representation of real numb ers

Betaexpansions

Denitions

The shift

Classes of numb ers

U representations

Denitions

The set of normal U representations

Normalization in a canonical linear numeration system

Representation of complex numb ers

Gaussian integers

Representability of the complex plane

Problems

Notes

Bibliography

Intro duction

This chapter deals with p ositional numeration systems Numb ers are seen as

nite or innite words over an alphab et of digits A numeration system is dened

by a couple comp osed of a base or a sequence of numb ers and of an alphab et

of digits In this chapter we study the representation of natural numb ers of

real numb ers and of complex numb ers We will present several generalizations

of the usual notion of numeration system which lead to interesting problems

Prop erties of words representing numb ers are well studied in numb er theory

the concepts of p erio d digit frequency normality give way to imp ortant results

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Numeration systems

Cantor sets can b e dened by digital expansions

In computer arithmetic it is recognized that algorithmic p ossibilities dep end

on the representation of numb ers For instance addition of two integers repre

sented in the usual binary system with digits and takes a time prop ortional

to the size of the data But if these numb ers are represented with signed digits

and then addition can b e realized in parallel in a time indep endent of

the size of the data

Since numb ers are words nite state automata are relevant to ols to describ e

sets of numb er representations and also to characterize the complexity of arith

metic op erations For instance addition in the usual binary system is a function

computable by a nite automaton but multiplication is not

Usual numeration systems such that the binary and the decimal ones are

describ ed in the rst section In fact these systems are a particular case of all

the various generalizations that will b e presented in the next sections

The second section is devoted to the study of the socalled b etaexpansions

intro duced by Renyi see Notes It consists in taking for base a real numb er

When is actually an integer we get the standard representation When

is not an integer a numb er may have several dierent representations A

particular representation playing an imp ortant role is obtained by a greedy

algorithm and is called the expansion it is the greatest in the lexicographic

order The set of expansions of numb ers of is shiftinvariant and its

closure called the shift is a symb olic dynamical system We give several

results on these topics We do not cover the whole eld which is very lively and

still growing It has interesting connections with numb er theory and symb olic

dynamics

In the third section we consider the representation of integers with resp ect

to a sequence of integers which can b e seen as a generalization of the notion

of base The most p opular example is the one of Fib onacci numb ers Every

p ositive integer can b e represented in such a system with digits and This

eld is closely related to the theory of b etaexpansions

The last section is devoted to complex numb ers Representing complex num

b ers as strings of digits allows to handle them without separating real and imag

inary part We show that every complex numb er has a representation in base

n i where n is an integer with digits in f n g This numeration

system enjoys prop erties similar to those of the standard ary system

For notations concerning automata and words the reader may want to con

sult Chapter

Standard representation of numb ers

In this section we will study standard numeration systems where the base is

a natural numb er We will represent rst the natural numb ers and then the

nonnegative real numb ers The notation intro duced in this section will b e used

in the other sections

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Standard representation of numb ers

Representation of integers

Let b e an integer called the base The usual ary representation of an

integer N is a nite word d d over the digit alphab et A f g

k 0

and such that

d N

i=0

Such a representation is unique with the condition that d This represen

tation is called normal and is denoted by

hN i d d

k 0

most signicant digit rst

The set of all the representations of the p ositive integers is equal to A

Let us consider the addition of two integers represented in the ary system

Let d d and c c b e two ary representations of resp ectively N and

k 0 k 0

M It is not a restriction to supp ose that the two representations have the

same length since the shortest one can b e padded to the left by enough zero es

Let us form a new word a a with a d c for i k Obviously

k 0 i i i

a N M but the a s b elong to the set f g So the

i i

i=0

word a a has to b e transformed into an equivalent one ie having the

k 0

same numerical value b elonging to A

More generally let C b e a nite alphab et of integers which can b e p ositive

or negative The numerical value in base on C is the function

C Z

c The normalization on which maps a word w c c of C onto

i n 0

i=0

C is the partial function

C A

that maps a word w c c of C such that N w is nonnegative onto

n 0

its normal representation hN i Our aim is to prove that the normalization is

computable by a nite transducer We rst prove a lemma

Lemma Let C b e an alphab et containing A There exists a right sub

sequential transducer that maps a word w of C such that N w onto

a word v b elonging to A and such that v N

Proof Let m maxfjc aj j c C a Ag and let m First

observe that for s Zand c C by the Euclidean division there exist unique

a A and s Z such that s c s a Furthermore if jsj then

js j jsj jc aj m

Consider the subsequential nite transducer A over C A where

A Q E is dened as follows The set Q fs Zj jsj g is the set of

p ossible carries the set of edges is

E fs s j s c s ag

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/ /

}

Figure Right subsequential transducer realizing the conversion in

base from f g onto f g

Observe that the edges are lettertoletter The terminal function is dened

by s hsi for s Q such that s

c Setting s there is a Now let w c c C and N

i 0 n 0

i=0

unique path

c a c a c a c a c a

2 2 n n 1 1 n1 n1 0 0

s s s s s

n+1 2 n 1 0

n n+1

By construction N a a a s hence the word v

0 1 n n+1

s a a has the same numerical value in base as w

n+1 n 0

Remark that v is equal to the normal representation of N if and only if it

do es not b egin with zero es

Example Figure gives the right subsequential transducer realizing

the conversion in base from the alphab et f g onto f g The signed

digit is denoted by

The two following results are a direct consequence of Lemma

Proposition In base for every alphab et C of p ositive integers con

taining A the normalization restricted to the domain C n C is a right sub

sequential function

Removing the zero es at the b eginning of a word can b e realized by a left

sequential transducer so the following prop erty holds true for any alphab et

Proposition In base for every alphab et C containing A the nor

malization on C is computable by a nite transducer

Corollary In base addition and subtraction with p ossibly zero es

ahead are right subsequential functions

Proof Take in Lemma C f g for addition and C

f g for subtraction

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Standard representation of numb ers

One proves easily that multiplication by a xed integer is a right subsequen

tial function and that division by a xed integer is a left subsequential function

see the Problems Section On the other hand the following result shows that

the p ower of functions computable by nite transducers is quite reduced

Proposition In base multiplication is not computable by a nite

transducer

Proof It is enough to show that the squaring function A A which

maps hN i onto hN i is not computable by a nite transducer Take for

n n n 2n n+1

instance and consider h i Then h i

2 2

n1 n n

Thus the image by of the set f j n g which is recognizable by

n1 n

a nite automaton is the set f j n g which is not recognizable thus

cannot b e computed by a nite transducer

Representation of real numb ers

Let b e an integer and set A f g A ary representation of

a nonnegative real numb er x is an innite sequence x of A such that

i ik

x x

This representation is unique and said to b e normal if it do es not end by

and if x when x It is traditionally denoted by

hxi x x x x

k 0 1 2

If x then there exists some i such that x We then put

x x The set of ary expansions of numb ers is equal to

1 i+1

N N

A n A n A the one of numb ers of is A n A The

set A is the set of all ary representations not necessarily normal

The word x x is the integer part of x and the innite word x x is

k 0 1 2

the fractional part of x Note that the natural numb ers are exactly those having

a zero fractional part compare with the representation of complex numb ers in

k +1

If hxi x x x x then x and by shifting we obtain

k 0 1 2

that

k +1

hx i x x x x

k 0 1 2

thus from now on we consider only numb ers from the interval When

x we will change our notation for indices and denote hxi x

i i1

Let C b e a nite alphab et of integers which can b e p ositive or negative

The numerical value in base on C is the function

C R

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Numeration systems

N i

which maps a word w c of C onto c The normalization on

i i1 i

C is the partial function

N N

C A

that maps a word w c such that x w b elongs to onto its

i i1

ary expansion hxi A n A

Proposition For every alphab et C containing A the normalization on

C is computable by a nite transducer

Proof First we construct a nite transducer B where edges are the reverse

of the edges of the transducer A dened in the pro of of Lemma Let

B Q F Q with set of edges

ca ca

F ft s j s t E g

Every state is terminal

Let

c a c a c a c a

n n 3 3 2 2 1 1

s s s s

n 2 1 0

b e a path in B starting in s Then

c a a s c

n 1 n n 1

n n n

Since A is sequential the automaton B is unambiguous that is given an input

word c C there is a unique innite path in B starting in and lab elled

i i1

P P

i i N

a b ecause for c by c a in C A and such that

i i i i i1

i1 i1

each n js j

To end the pro of it remains to show that the function which given a word in

A transforms it into an equivalent word not ending by is computable

N N

by a nite transducer and this is clear from the fact that A A n A

N N

is a rational subset of A A see Chapter

Corollary Additionsubtraction multiplicationdivision by a xed

integer of real numb ers in base are computable by a nite transducer

Example Figure gives the nite transducer realizing non normal

ized addition meaning that the result can end by the improp er sux of

real numb ers on the interval in base

Betaexpansions

We now consider numeration systems where the base is a real numb er

Representations of real numb ers in such systems were intro duced by Renyi

under the name of expansions They arise from the orbits of a piecewise

monotone transformation of the unit interval T x x mo d see b elow

Such transformations were extensively studied in ergo dic theory and symb olic

dynamics

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Betaexpansions

/ /

}

? ?

Figure Finite transducer realizing non normalized addition of real

numb ers in base

Denitions

Let the base b e a real numb er Let x b e a real numb er in the interval

A representation in base or a representation of x is an innite word

x such that

i i1

x x

A particular representation called the expansion can b e computed

by the greedy algorithm denote by by c and fy g the integer part and the

fractional part of a numb er y Set r x and let for i x b r c

0 i i1

x r f r g Then x

i i i1

The expansion of x will b e denoted by d x

An equivalent denition is obtained by using the transformation of the

unit interval which is the mapping

T x x mo d

Then d x x if and only if x b T xc

i i1 i

Let x b e any real numb er greater than There exists k N such that

k k +1 k +1

x Hence x thus it is enough to represent numb ers

from the interval since by shifting we will get the representation of any

p ositive real numb er

p p

Example Let b e the golden ratio For x we

have d x

If is not an integer the digits x obtained by the greedy algorithm are

elements of the alphab et A f b cg called the canonical alphabet

When is an integer the expansion of a numb er x of is exactly

the standard ary expansion ie d x hxi and the digits x b elong to

f g However for x there is a dierence hi but d

As we shall see later the expansion of plays a key role in this theory

Another characterization of a expansion is the following one

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Lemma An innite sequence of nonnegative integers x is the

i i1

expansion of a real numb er x of resp of if and only if for every i

i i1 i+1

resp i x x

i i+1

Proof Let x and let d x x By construction for i

i i1

r x x thus the result follows

i1 i i1

A real numb er may have several representations However the expan

sion obtained by the greedy algorithm is characterized by the following prop

erty

Proposition The expansion of a real numb er x of is the greatest

of all the representations of x with resp ect to the lexicographic order

Proof Let d x x and let s b e another representation of

i i1 i i1

x Supp ose that x s then there exists k such that x

i i1 i i1 k

P P

i i

s one gets x s and x x s s From

i i k 1 k 1 1 k 1

ik ik

P P

i i k

s which is imp ossible since by Lemma x

i i

ik +1 ik +1

i k

ik +1

Example continued Let b e the golden ratio The expansion of

x is equal to Dierent representations of x are or

for instance

As in the usual numeration systems the order b etween real numb ers is given

by the lexicographic order on expansions

Proposition Let x and y b e two real numb ers from Then x y

if and only if d x d y

Proof Let d x x and let d y y and supp ose that d x

i i1 i i1

d y There exists k such that x y and x x y y Hence

k k 1 k 1 1 k 1

1 k +1 k k 1 k 2

x y y y x x y

1 k 1 k k +1 k +2

k 1 k 2 k

since x x The converse is immediate

k +1 k +2

If a representation ends in innitely many zeros like v the ending zeros

are omitted and the representation is said to b e nite Remark that the

x for some i and it is expansion of x is nite if and only if T

eventually p erio dic if and only if the set fT x j i g is nite Numb ers

such that d is eventually p erio dic are called numbers and those such that

d is nite are called simple numb ers

Remark The expansion of is never purely p erio dic

Indeed supp ose that d is purely p erio dic d a a with n

1 n

1 n n

minimal a A Then a a which means that

i 1 n

a a a is a representation of and a a a d

1 n1 n 1 n1 n

which is imp ossible

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Betaexpansions

The expansion of Example Let b e the golden ratio

is nite equal to d

Let The expansion of is eventually p erio dic equal to

Let Then d We shall see later that it is

ap erio dic

The shift

Recall that the set A is endowed with the lexicographic order the pro duct

top ology and the onesided shift dened by x x Denote

i i1 i+1 i1

by D the set of expansions of numb ers of It is a shiftinvariant subset

N N

of A The shift S is the closure of D and it is a subshift of A When

is an integer S is the full shift A

The greedy algorithm computing the expansion can b e rephrased as fol

lows

Lemma The identity

d T d

holds on the interval

x and Proof Let x and let d x x Then T x

i i i1

the result follows

In the case where the expansion of is nite there is a sp ecial repre

sentation playing an imp ortant role Let us intro duce the following notation

d if d is innite and d Let d t and set d

i i1

t t t if d t t t is nite

1 m1 m 1 m1 m

When is an integer representations ending by the innite word d

are the improp er representations

Example Let then d and d

and d d For

The set D is characterized by the expansion of as shown by the following

result b elow Notice that the sets of nite factors of D and of S are the same

and that d is the supremum of S but that in case d is nite d is

not an element of S

Theorem Let b e a real numb er and let s b e an innite sequence

of nonnegative integers The sequence s b elongs to D if and only if for all

s d

and s b elongs to S if and only if for all p

s d

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Numeration systems

Proof First supp ose that s s b elongs to D then there exists x in

i i1

such that s d x By Lemma for every p d x d T x

Since T x and d is a strictly increasing function Prop osition

p p

d x s d

In the case where d t t is nite supp ose there exists a p such

1 m

p p

that s d Since s d we get s t s t

p+1 1 p+m1 m1

s t Iterating this pro cess we see that s d which do es

p+m m

not b elong to D a contradiction

Conversely let d d and supp ose that for all p s d

i i1

By induction let us show that for all r for all i

s d d s

p+r i+1 i+r p+1

s s d d

p+1 p+r i+1 i+r

r r

This is obviously satised for r

Supp ose that s s d d

p+1 p+r +1 i+1 i+r +1

First assume that s d then s s d d By

p+1 i+1 p+2 p+r +1 i+2 i+r +1

induction hyp othesis

s s d d

p+2 p+r +1 i+2 i+r +1

2 r +1 2 r +1

and the result follows

then Next supp ose that s d Since for all p s d

p+1 i+1

s s d d thus

p+2 p+r +1 1 r

s d d d d s

p+r +1 i+1 1 r i+1 p+1

r +1 2 r +1

2 r +1

since d d

1 r

Thus for all p for all i

X X

r r

s d

p+r i+r

r 1 r 1

P P

r r

d if is not an s In particular for i

r +1 p+r

r 1 r 1

integer and the result follows by Lemma

If is an integer then d If for all p s d

then every letter of s is smaller than or equal to and s do es not end by

therefore s b elongs to D

D For the shift we have the following situation A sequence s b elongs to

(n)

if and only if for each n there exists a word v of D such that s s is

1 n

(n)

a prex of v Hence s b elongs to S if and only if for every p for every

p p

n s s d or equivalently if s d

1 n

From this result follows the following characterization a sequence is the

expansion of for a certain numb er if and only if it is greater than all its

shifted sequences

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Betaexpansions

Corollary Let s s b e a sequence of nonnegative integers

i i1

with s and for i s s and which is dierent from Then there

1 i 1

exists a unique real numb er such that s Furthermore s is

the expansion of if and only if for every n s s

s z and denote by Proof Let f b e the formal series dened by f z

its radius of convergence Since s s we get s Since

i 1 1

for z the function f is continuous and increasing and since f

and f z for z sucient close to it follows that the equation f z

has a unique solution If exists such that f we get that

s f s thus must b e b etween s and s On the

1 1 1 1

other hand f s s s If s f s If s and if

1 1 1 1 1 1

the s s are eventually then f s otherwise lim f z Thus

i 1 z 1

in any case there exists a real s s such that f

1 1

Now we make the following hyp othesis H for all n s s Supp ose

that the expansion of is d t s Since s is a representation of

s t Hence for each n s s d If d is innite by

Theorem s b elongs to D a contradiction

If d is nite say d t t either s d and as ab ove we get

1 m

s d In fact s cannot b e purely p erio dic that s is in D or d

Thus s is necessarily b ecause of hyp othesis H thus it is dierent from d

of the form t t t t t for some k So s t

1 m1 m 1 m k m+1 1

k m

s t and s s b ecause s t contradicting hyp othesis

k m+m m m m

H Hence the expansion of is s

Conversely supp ose that s d for some From Theorem

If d is If d is innite d d for every n s d

d nite d

Let us recall some denitions on symb olic dynamical systems or subshifts

N +

see Chapter Section Let S A b e a subshift and let I S A n F S

b e the set of factors avoided by S Denote by X S the set of words of I S

which have no prop er factor in I S The subshift S is of nite type i the set

X S is nite The subshift S is soc i X S is a rational set It is equivalent

to say that F S is recognized by a nite automaton The subshift S is said

to b e coded if there exists a prex co de Y A such that F S F Y or

equivalently if S is the closure of Y

To the shift a prex co de Y Y is asso ciated as follows It is the set

of words which for each length are strictly smaller than the prex of d of

same length more precisely if d t is innite set Y ft t a j

i i1 1 n1

a t n g with the convention that if n t t If

n 1 n1

d t t let Y ft t a j a t n mg

1 m 1 n1 n

Proposition The shift S is co ded by the co de Y

Proof First if d t is innite let us show that D Y Let s D

i i1

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Numeration systems

v By Theorem s d thus can b e written as s t t a

1 1 n 1 n

1 1

and v d Iterating this pro cess we see that s t with a

1 n n

1 1

t a Y Conversely let s u u Y with u t t a

n n 1 2 i 1 n 1 n

i i i i

Then s d For each p s b egins with a word of the form

p j 1

t t b with b t thus s d d t

j +1 j +r 1 j +r j +r j +r j

p p p p p p

Next if d t t is nite we claim that Y S First let s S

1 m

t v with n m a By Theorem s d thus s t t a

n 1 1 n 1 n 1 n

1 1 1 1

and v d Iterating the pro cess we get s S Conversely let s Y

n m As ab ove one gets that for each s u u with u t t a

i 1 2 i 1 n 1 n

i i

p s d

We now compute the top ological entropy of the shift

hS log

F (S )

see for denitions and notations In the case where the shift is soc by

Theorem the entropy hS can b e shown to b e equal to log We show

b elow that the same result holds true for any kind of shift

Proposition The top ological entropy of the shift is equal to log

Proof For n the numb er of words of length n of Y is clearly equal to t

thus the generating series of Y is equal to

f z t z

Y n

By Corollary is the unique p ositive solution of f z Since Y is

It is thus enough to show that a co de by Lemma

Y Y

F (S )

Let p b e the numb er of factors of length n of the elements of S and let

f p z

F (S )

Let c b e the numb er of words of length n of Y and let

f c z

F (Y )

Since any word of Y is in F S we have c p On the other hand let

n n

w b e a word of length n in F S By Prop osition w can b e uniquely

written as w u t t where u Y ju j n i and i n Thus

i 1 i i i

have the same radius of p c c Hence the series f and f

n n 0 Y

F (S )

convergence and the result is proved

We now show that the nature of the subshift as a symb olic dynamical system

is entirely determined by the expansion of

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Betaexpansions

Theorem The shift S is soc if and only if d is eventually

p erio dic

Proof Supp ose that d is innite eventually p erio dic

d t t t t

1 N N +1 N +p

with N and p minimal We use the classical construction of minimal nite

automata by right congruent classes see Chapter Let F D b e the set of

nite factors of D We construct an automaton A with N p states q

q where q i represents the right class t t and q stands

N +p i 1 i1 1

F (D )

for For each i i N p there is an edge lab elled t from q to

i i

F (D )

q There is an edge lab elled t from q to q For i N p

i+1 N +p N +p N +1

there are edges lab elled by t from q to q Let q b e the only initial

i i 1 1

state and all states b e terminal That F D is precisely the set recognized

by the automaton A follows from Theorem Remark that when the

expansion of happ ens to b e nite say d t t the same construction

1 m

applies with N m p and all edges from q lab elled by t

m m

leading to q

Supp ose now that d t is not eventually p erio dic nor nite There

i i1

exists an innite sequence of indexes i i i such that the sequences

1 2 3

t t b e all dierent for all k Thus for all pairs i i j t

i +1 i +2 j i

k k k

t there exists p such that for instance t t and t

i +p1 i +p i +p i

j j j

t w with the convention that when p w We have t

i +p1 i

that t t w t F D t t w t F D t t w t

1 i 1 i +p 1 i 1 i +p 1 i 1 i +p

j j j

and F D but t t w t do es not b elong to F D Hence t t

1 i 1 i +p 1 i

j j

are not right congruent mo dulo F D The numb er of right congruence t t

1 i

classes is thus innite and F D is not recognizable by a nite automaton

Example For d and the shift is soc

We have a similar result when the expansion of is nite

Theorem The shift S is of nite typ e if and only if d is nite

Proof Let us supp ose that d t t is nite and let

1 m

i m

Z fu A j u t t g fu A j ut t g

1 i 1 m

2im1

Clearly Z A n F S The set X S of words forbidden in S which are

minimal for the factor order is a subset of Z Since Z is nite X S is nite

and thus S is of nite typ e

Conversely supp ose that the shift is of nite typ e It is thus soc and

by Theorem d is eventually p erio dic Supp ose that d is not

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Numeration systems

nite d t t t t with N and p minimal and

1 N N +1 N +p

t t Let

N +1 N +p

Z f t t t h j j N h t t g

1 j 1 j j j 1 j

f t t t t t t t h

1 N N +1 N +p N +1 N +j 1 N +j N +j

j k j p h t t g

N +j 1 N +j

Clearly Z A n F S

Case Supp ose there exists j p such that t t and t t

j N +j 1 N +1

(k ) k

t t For k xed let w t t t t t t Z

j 1 N +j 1 1 N N +1 N +p 1 j

We have t t t t t t F S On the other hand

1 N N +1 N +p N +1 N +j 1

for n t t t t is strictly smaller in the lexicographic order

n N N +1 N +p

than the prex of d of same length the inequality is strict since the t s

are not all equal for i N p thus t t t t t t

n N N +1 N +p 1 j

(k )

F S Hence any strict factor of w is in F S Therefore for any k

(k )

w X S and X S is thus innite the shift is not of nite typ e

Case No such j exists then d t t which is imp ossible by

1 N

Remark

Example For the shift is of nite typ e it is the

golden mean shift describ ed in Example

Classes of numb ers

Recall that an algebraic integer is a ro ot of a monic p olynomial with integral

co ecients An algebraic integer is called a Pisot number if all its Galois

conjugates have mo dulus less than one It is a Salem number if all its conjugates

have mo dulus and at least one conjugate has mo dulus one It is a Perron

number if all its conjugates have mo dulus less than

Example Every integer is a Pisot numb er The golden ratio

p p

and its square are Pisot numb ers with minimal p olynomial

2 2

resp ectively X X and X X

A rational numb er which is not an integer is never an algebraic integer

is a Perron numb er which is neither Pisot nor Salem

The most imp ortant result linking shifts and numb ers is the following one

Theorem If is a Pisot numb er then the shift S is soc

This result is a consequence of a more general result on expansions of

numb ers of the eld Q when is a Pisot numb er It is a partial generalization

of the well known fact that when is an integer numb ers having an eventually

p erio dic expansion are the rational numb ers of see Problems Section

Proposition If is a Pisot numb er then every numb er of Q

has an eventually p erio dic expansion

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Betaexpansions

d d1

Proof Let P X X a X a b e the minimal p olynomial of

1 d

and denote by the conjugates of Let x b e arbitrarily xed

1 2 d

in Q It can b e expressed as

i 1

p x q

i=0

with q and p in Z q as small as p ossible in order to have uniqueness

Let x b e the expansion of x and denote by

k k 1

x x

n+2 n+1

n k n (1)

x x T x r r r x

k n n

k =1

For j d let

n d1

X X

i (j ) (j ) n 1

p x r r x q

i k

j n n j

i=0

k =1

Let max j j since is a Pisot numb er Since x b c we get

2j d j k

n1 d1

X X

k (j ) 1 n+i

jr j q jp j b c

i=0

k =0

(j )

and since max sup jr j

1j d

(1) (d)

We need a technical result Set R r r and let B b e the matrix

n n

1ij d

i 1

p For every n there exists a unique Lemma Let x q

i=0

(1) (d)

d 1

duple Z z z in Z such that R q Z B

n n

n n n

(1)

Proof By induction on n First r r x x thus

1 1

(1) (d)

z z

i+1 1 1 1 1

p q x q r q

i 1 1

i=0

(1)

d d1

using the fact that a a a Z Now r r

1 d j n+1

n+1

r x hence

n n+1

(1) (d)

(d) (2)

z z

n n

n+1 n+1

1 1 (1)

q x q r q z

n+1 n+1

d1 d

(1)

q x Z since z

n+1

Thus

d n d1

X X X

k (k ) k 1 i n 1 (1)

z x q p q r r

k i n

n n

i=0

k =1 k =1

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Numeration systems

Since the latter equation has integral co ecients and is satised by it is also

satised by each conjugate j d

n d d1

X X X

k k

i 1 (j ) n 1 (k )

p x q r q z

i k

j n j n

j j

i=0

k =1 k =1

We resume the pro of of Prop osition Let V q R The V

n n n n1

(j )

j As the matrix B is have b ounded norm since max sup jr

1j d

invertible for every n

(1) (d) (j )

jjZ jj jjz z jj max jz j

n n n

1j d

so there exist p and m such that Z Z hence r r and the

m+p m m+p m

expansion of x is eventually p erio dic

On the other hand there is a gap b etween Pisot and Perron numb ers as

shown b e the following result

Proposition If S is soc then is a Perron numb er

Proof With the automaton A dened in the pro of of Theorem one

asso ciates a matrix M M by taking for M i j the numb er of edges from

state q to state q that is if d t t t t

i j 1 N N +1 N +p

M i t

M i i for i N p

M N p N

and other entries are equal to

N +p N +p

Claim The matrix M is primitive M since M i j is equal to

the numb er of paths of length N p from q to q in the strongly connected

i j

automaton A

Claim The characteristic p olynomial of M is equal to

N +p

X X

N i N +pi N N +p

t X t X X K X X

i i

i=1 i=1

and is one of its ro ots it can b e checked by a straightforward computation

When d t t is nite the matrix asso ciated with the automaton is

1 m

m m1

simpler it is the companion matrix of the p olynomial K X X t X

t which is primitive since M

Since is an eigenvalue of a primitive matrix by the theorem of Perron

Frob enius is strictly greater in mo dulus than its algebraic conjugates

Thus when is a nonintegral rational numb er for instance the shift

S cannot b e soc

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U representations

Example There are Perron numb ers which are neither Pisot nor Sa

lem numb ers and such that the shift is of nite typ e for instance the ro ot

4 3 2

of X X X satises d and has a conjugate

Remark If is a Perron numb er with a real conjugate then d

cannot b e eventually p erio dic

In fact supp ose that d t t t t and that has a conju

1 N N +1 N +p

gate Since is a zero of the p olynomial K X of ZX is also a zero

of this p olynomial Thus d d and by Corollary

has a real conjugate For instance the quadratic Perron numb er

and thus S is not soc

U representations

We now consider another generalization of the notion of numeration system

which only allow to represent the natural numb ers The base is replaced by an

innite sequence of integers The basic example is the wellknown Fib onacci

numeration system

Denitions

Let U u b e a strictly increasing sequence of integers with u A

n n0 0

representation in the system U or a U representation of a nonnegative

integer N is a nite sequence of integers d such that

i k i0

d u N

i i

i=0

Such a representation will b e written d d most signicant digit rst

k 0

Among all p ossible U representations of a given nonnegative integer N one

is distinguished and called the normal U representation of N it is sometimes

called the greedy representation since it can b e obtained by the following greedy

algorithm given integers m and p let us denote by q m p and r m p the

quotient and the remainder of the Euclidean division of m by p Let k such

that u N u and let d q N u and r r N u and for i k

k k +1 k k k k

d q r u and r r r u Then N d u d u The

i i+1 i i i+1 i k k 0 0

normal U representation of N is denoted by hN i

By convention the normal representation of is the empty word Under

the hyp othesis that the ratio u u is b ounded by a constant as n tends

n+1 n

to innity the integers of the normal U representation of any integer N are

b ounded and contained in a canonical nite alphab et A asso ciated with U

Example Let U f j n g The normal U representation of an

integer is nothing else than its ary standard expansion

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Numeration systems

Example Let F F b e the sequence of Fib onacci numb ers see

n n0

Example The canonical alphab et is equal to A f g The normal

F representation of the numb er is another representation is

An equivalent denition of the notion of normal U representation is the

following one

Lemma The word d d where each d for k i is a nonnega

k 0 i

tive integer and d is the normal U representation of some integer if and

only if for each i d u d u u

i i 0 0 i+1

Proof If d d is obtained by the greedy algorithm r d u d u

k 0 i+1 i i 0 0

u by construction

i+1

As for expansions the U representation obtained by the greedy algorithm

is the greatest one for some order we dene now Let v and w b e two words

We say that v w if jv j jw j or if jv j jw j and there exist letters a b such

that v uav and w ubw This order is sometimes called radix order or

genealogic order or even lexicographic order in the literature although the

denition is slightly dierent from the usual denition of lexicographic order on

nite words see Chapter

Proposition The normal U representation of an integer is the greatest

in the radix order of all the U representations of that integer

Proof Let d d d b e the normal U representation of N and let w

k 0

w w b e another representation Since u N u k j If k j

j 0 k k +1

then d w If k j supp ose d w Thus there exists i k i

such that d w and d d w w Hence d u d u

i i k i+1 k i+1 i i 0 0

w u w u but d u d u w u d u d u

i i 0 0 i i 0 0 i i i1 i1 0 0

so u w u w u d u d u u since d is normal

i i1 i1 0 0 i1 i1 0 0 i

which is absurd

The order b etween natural numb ers is given by their radix order b etween

their normal U representations

Proposition Let M and N b e two nonnegative integers then M N

if and only if hM i hN i

U U

Proof Let v v v hM i with u M u and w w w

k 0 U k k +1 j 0

hN i with u N u and supp ose that v w Then k j If k j

U j j +1

u u and M N If k j there exists i such that v w and

k +1 j i i

v v w w Hence

k i+1 k i+1

M v u v u

k k 0 0

w u w u w u v u v u

k k i+1 i+1 i i i1 i1 0 0

w u w u w u N

k k i+1 i+1 i i

since v u v u u by Lemma thus M N

i1 i1 0 0 i

Version April

U representations

The set of normal U representations

The set of normal U representations of all the nonnegative integers is denoted

by LU

Example continued Let F b e the sequence of Fib onacci numb ers The

set LF is the set of words without the factor and not b eginning with a

LF f g n f g f g

First the analogue of Theorem is the following result

Proposition The set LU is the set of words over A such that each

sux of length n is less in the radix order than hu i

n U

Proof Let v v v b e in LU and n k By Lemma

k 0

v u v u u and by Prop osition v v hu i

n1 n1 0 0 n n1 0 n U

The converse is immediate

An imp ortant case is when LU is recognizable by a nite automaton as it

is the case for usual numeration systems We rst give a necessary condition

Recall that a formal series with co ecients in N is said to b e Nrational if it

b elongs to the smallest class containing p olynomial with co ecients in N and

closed under addition multiplication and star op eration where F is the series

2 n

F F F F F b eing a series such that F A

Nrational series is necessarily Zrational and thus can b e written P X QX

with P X and QX in ZX and Q Therefore the sequence of co e

cients of a Nrational series satises a linear recurrent relation with co ecients

in Z It is classical that if L is recognizable by a nite automaton then the

series f X X where denotes the numb er of words of length n

L n n

in L is Nrational see Berstel and Reutenauer

Proposition If the set LU is recognizable by a nite automaton

u X is Nrational and thus the sequence U then the series U X

satises a linear recurrence with integral co ecients

Proof Let b e the numb er of words of length n in LU The series f X

L(U )

X is Nrational We have u b ecause the numb er of

n n n 0

words of length n in LU is equal to the numb er of naturals smaller than u

whose normal representation has length n Thus U X f X X

L(U )

and it is Nrational

When the sequence U satises a linear recurrence with integral co ecients

we say that U denes a linear numeration system

To determine sucient conditions on the sequence U for the set LU to b e

recognizable by a nite automaton is a dicult question see Problem It

is strongly related to the theory of expansions where is the dominant ro ot of

the characteristic p olynomial of the linear recurrence of U Nevertheless there

is a case where the set LU and the factors of the shift coincide This means

Version April

Numeration systems

that the dynamical systems generated by the expansions of real numb ers and

by normal U representations of integers are the same

It is obvious that if a word of the form v b elongs to LU then v itself is

a word of LU but the converse is not true in general We will say that a set

L A is rightextendable if the following prop erty holds

v L v L

Theorem Let U u b e a strictly increasing sequence of integers

n n0

with u and such that sup u u and let A b e the canonical

0 n+1 n

alphab et There exists a real numb er such that LU F D if and

d the sequence only if LU is rightextendable In that case if d

i i1

U is determined by

u d u d u

n 1 n1 n 0

Proof Clearly if LU F D for some then LU is rightextendable

Conversely supp ose that LU is rightextendable For each n denote

(n)

d hu i d

n U

(k ) (k )

LU and Since LU is rightextendable for each k n d d

(n) (n) (k ) (k ) (n) (n) (k ) (k )

b ecause d d Therefore d d d d thus d d

1 1 1 1

k k k k

(k ) (k )

is the greatest word of length k in the radix order d d

(n)

then d d d d Let d d If there exists m Let d d

n n+1 1 2 i i1 n

such that d d then d is p erio dic Let m b e the smallest such index In

that case put t d t d t d t for i m In

1 1 m1 m1 m m i

case d is not p erio dic put t d for every i Then the sequence t satises

i i i i1

t t t t for all n and thus by Corollary there exists a

n n+1 1 2

unique such that d t

i i1

Let us show that LU F D Recall that

D fs j p s d d g

i i1

hence

F D fv v v j n n k v v d d hu i g

k 0 n1 0 1 n n U

by Prop osition

Now since by denition d d hu i we get

1 n n U

u d u d u

n 1 n1 n 0

The numeration systems satisfying Theorem will b e called canonical

numeration systems asso ciated with and denoted by U Note that if d

is eventually p erio dic then LU is recognizable by a nite automaton and U

satises a linear recurrent sequence

Example continued The Fib onacci numeration system is the canoni

cal numeration system asso ciated with the golden ratio

Version April

U representations

Normalization in a canonical linear numeration system

We rst give general denitions valid for any linear numeration system dened

by a sequence U The numerical value in the system U of a representation

w d d is equal to w d u Let C b e a nite alphab et of

k 0 U i i

i=0

integers The normalization in the system U on C is the partial function

C A

that maps a word w of C such that w is nonnegative onto the normal

U representation of w

In the sequel we assume that U U is the canonical numeration system

asso ciated with a numb er which is a Pisot numb er Thus U satises an

equation of the form

u a u a u a u a Z a n m

n 1 n1 2 n2 m nm i m

In that case the canonical alphab et A asso ciated with U is A f K g

m m1

where K maxu u The p olynomial P X X a X a

i+1 i 1 m

will b e called the characteristic polynomial of U

We also make the hyp othesis that P is exactly the minimal p olynomial of

in general P is a multiple of the minimal p olynomial

Our aim is to prove the following result

Theorem Let U U b e a canonical linear numeration system asso ci

ated with a Pisot numb er and such that the characteristic p olynomial of U

is equal to the minimal p olynomial of Then for every alphab et C of nonneg

ative integers the normalization on C is computable by a nite transducer

The pro of is in several steps Let C f cg C fc cg and let

d u g Z U c fd d j d C

i i k 0 i

i=0

b e the set of words on C having numerical value in the system U We rst

prove a general result

Proposition If Z U c and LU are recognizable by a nite automa

ton then is a function computable by a nite transducer

Proof Let f f f and g g g b e two words of C with for instance

n 0 k 0

n k We denote by f g the word of C equal to f f f g f

n k +1 k k 0

g The graph of is equal to c ff g C A j g LU f g

0 C C

Z U cg

Let R b e the graph of

R a a a a a b a b

aC aC abC

Version April

Numeration systems

R is a rational subset of C C C Let us consider the set

R R C LU Z U c C A C

Then c is the pro jection of R on C A As LU and Z U c are rational by

assumption C LU Z U c is a recognizable subset of C A C as

a Cartesian pro duct of rational sets see Berstel Since R is rational R

is a rational subset of C A C So c b eing the pro jection of R c is a

C C

rational subset of C A that is is computable by a nite transducer

The core of the pro of relies in the following result

Proposition Let U b e a linear numeration system such that its char

acteristic p olynomial is equal to the minimal p olynomial of a Pisot numb er

Then Z U c is recognizable by a nite automaton

Proof Set Z Z U c for short We dene on the set H of prexes of Z the

equivalence relation as follows m is the degree of P

n n

f g n n m f g

U U

n n

Let f g It is clear that the sequences f and g satisfy

U n0 U n0

the same recurrence relation as U Since they coincide on the rst m values

they are equal Thus for any h C

jhj

f h Z f h

U U

jhj

g h

U U

g h Z

which means that f and g are right congruent mo dulo Z If f and g are not in

H then f g as well

It remains to prove that has nite index This will b e achieved by showing

that there are only nitely many p ossible values of f for f H and for

all n m Recall that if are the ro ots of P since

1 2 m

P is minimal they are all distinct and there exist complex constants

such that for all n N

2 m

i n

i=1

If f f f let f f f f

k 0 k 1 0

Claim There exists such that for all f C

j f f j

U 1

Version April

U representations

We have

k k

X X

f u f f f

j j 1 j U 1

j =0 j =0

k m k

X X X

f f

j 1 i j

j =0 i=1 j =0

m k

X X

i j

i=2 j =0

Since is a Pisot numb er j j for i m and

j j j f f j c

i U 1

j j

i=2

Claim There exists such that for all f H j f j

Since f H there exists h C such that f h Z Thus

jhj jhj

f h f h

U U 1 1

jhj jhj jhj

f c

1 1

thus f c Similarly f c hence j f j

1 1

Claim There exists such that for all f H for all n m

j f j

We have

n n n n

j f j j f f j j f j

U U 1 1

j f j

n m1

hence j f j

U 1

Thus there are only nitely many p ossible values of f for f H and

for all n m therefore has nite index and Z U c is rational

Proof of the theorem Since U is canonical for a Pisot numb er LU is recog

nizable by a nite automaton The result follows from Prop osition and

Prop osition

Corollary Under the same hyp othesis as in Theorem addi

tion of integers represented in the canonical linear numeration system U is

computable by a nite transducer

Proof The canonical alphab et b eing A f K g take C f K g in

Theorem

Version April

Numeration systems

~ =

- -

> }

Figure Automaton recognizing the set of words on f g having

value in the Fib onacci numeration system

> }

? ?

Figure Normalization on f g in the Fib onacci numeration system

Example continued Let F b e the sequence of Fib onacci numb ers The

characteristic p olynomial of F is X X and it is the minimal p olynomial of

Figure gives the automaton recognizing the Pisot numb er

the set Z F of words on the alphab et f g having numerical value in

the Fib onacci numeration system

Figure shows a nite transducer realizing the normalization on f g in

the Fib onacci numeration system For simplicity we assume that input and

output words have the same length

The result stated in Theorem can b e extended to the case where U

is not the canonical numeration system asso ciated with a Pisot numb er but

where the characteristic p olynomial of U is still equal to the minimal p olynomial

of There is a partial converse to this result see Notes

Representation of complex numb ers

The usual metho d of representing real numb ers by their decimal or binary ex

pansions can b e generalized to complex numb ers It is p ossible see the Problem

Section to represent complex numb ers with an integral base and complex digits

but we present here results when the base is some complex numb er

Version April

Representation of complex numb ers

Gaussian integers

In this section we fo cus on representing complex numb ers using integral digits

The set of Gaussian integers denoted by Zi is the set fa bi j a b Zg The

base will b e chosen as a Gaussian integer It is quite natural to extend prop

erties satised by integral base for real numb ers namely the fact that integers

coincide with numb ers having a zero fractional part More precisely given a

base of mo dulus and an alphab et A of digits that are Gaussian integers

we will say that A is an integral numeration system for the eld of complex

numb ers C if every Gaussian integer z has a unique integer representation of

d with d A We shall see later the form d d such that z

j j k 0

j =0

that in that case every complex numb er has a representation

We rst show preliminary results A set A Zi is a complete residue system

for Zi mo dulo if every element of Zi is congruent mo dulo to a unique

2 2

element of A The norm of a Gaussian integer z x y i is N z x y

The following result is well known in elementary numb er theory

Theorem Gauss Let a bi b e a nonzero Gaussian integer and

let N b e the norm of If a and b are coprime then a complete residue system

for Zi mo dulo is the set

f N g

If gcda b a complete residue system for Zi mo dulo is the set

fp iq j p N q g

We use it in the following circumstances

Proposition Supp ose that every Gaussian integer has an integer rep

resentation in A Then this representation is unique if and only if A is a

complete residue system for Zi mo dulo that contains

Proof Let us supp ose that A is a complete residue system containing and

let d d and c c b e two representations of z in A One can supp ose

k 0 p 0

k 1 p1

d c Then c d d d c c thus d and

0 0 0 0 k 1 p 1 0

c are congruent mo dulo and are elements of A thus they are equal which

is absurd

Conversely supp ose that every Gaussian integer z has a unique representa

tion of the form d d with digits d in A Then z is congruent to d mo dulo

k 0 j 0

thus the digit set A must contain a complete residue system

Now let c and d b e two digits of A that are congruent mo dulo Then

c d q with q in Zi Let q q b e the representation of q Hence c has

n 0

two representations c itself and q q d

n 0

If we require the digits to b e natural numb ers the base must b e a Gaussian

integer a bi with a and b coprime and the choice is drastically restricted

Version April

Numeration systems

Theorem Let b e a Gaussian integer of norm N and let A f

N g Then A is an integral numeration system for the complex numb ers

if and only if n i for some n

2 2

Proof First let a bi a and b coprime and let A f a b g

Supp ose that a We shall show that the Gaussian integer z a ib

has no representation Supp ose in the contrary that z has a representation

2 2

d d Let y z a b a Since a y b elongs to A

k 0

k k +1

But y d d d d d d Thus y is congruent

0 1 0 k k 1 k

to d mo dulo and so y d It follows that d d d d

0 0 1 0 k k 1

d so for j k d Thus y and a b But is

k j

not the base of a numeration system

If a and b then i is not a base either If a and jbj

the digit set is f b g If b then i has no integer representation

since hii b b If b then i has no integer representation see

Exercise

Let now a and b Supp ose that a Gaussian integer z has a

representation d d Then Imz d Im d Im Since Im b is

k 0 k 1

a divisor of Im for all k b divides Imz Take z i Since b there is a

contradiction

Let now n i n and thus A f n g It remains to prove

that any z Zi has an integer representation in A Let z x iy x and

y in Z We have z c d with d y and c x ny From the equality

2 2 3 2

n n it is p ossible to write z as z d d d d

3 2 1 0

with d N

Let z d d with d N and k and let d d d N

k 0 i k 0

Denote by S the sumofdigits function

S C N N

z d S z d d d

k 0

2 3 2 2

In the following we will use the fact that n n n

2 2

that is hn i is equal to the word n n and that the sum

of digits of these two representations is the same and equal to n By

2 2 2

the Euclidean division by n d r q n with r n thus

0 0 0 0

2 2 3 4 k

z r d q n d q n d q d d

0 1 0 2 0 3 0 4 k

(1) (1) (1) (1)

k (1) (1)

d d Clearly S z d S z d where d d d

0 0

k k

(1) (1)

k 1 (1)

Let z d d then S z d S z d and the inequality

1 1

is strict if and only if r Rep eating this pro cess we get z z r

0 1 0

z z r z z r with for i j r A and

1 2 1 j 1 j j 1 i

(1) (j 1)

S z d S z d S z d

1 j 1

(j )

Since the sequence S z d of natural numb ers is decreasing there ex

j j

(p) (p+m) m

ists a p such that for every m S z d S z d thus

p p+m

divides z for every m therefore z So we get

p p

hz i r r

p1 0

Version April

Representation of complex numb ers

Let now n i Using the result for the conjugate n i we have

r r hzi

p1 0

for every Gaussian integer z Hence

hz i r r

p1 0

for every Gaussian integer z

From this result one can deduce that every complex numb er is representable

in this system

Theorem If n i n and A f n g every complex

numb er has a representation not necessarily unique in the numeration system

Proof Let z x iy x and y in R b e a xed arbitrary complex numb er For

k let u iv Then

k k

x iy u iv p iq r is

k k k k k k

k k k

where xu y v p r xv y u q s with p and q in Z and

k k k k k k k k k k

jr j js j Let

k k

p iq r is

k k k k

z y

k k

Since y when k lim z z Since p iq is a Gaussian integer

k k k k k

by Theorem

(k ) (k )

d hp iq i d

k k

t(k )

Thus

(k ) (k )

t(k )k k

d z d

t(k )

(k )

(k ) (k )

k 1

0 t(k )k

d j jz j jd

t(k )

j j j j

jz j jy j n

j j j j

c jz j jy j

j j

where c is a p ositive constant not dep ending on k

Since the representation of a Gaussian integer is unique and since Zi is a

discrete lattice ie is an additive subgroup such that any b ounded part contains

only a nite numb er of elements tk k has an upp er b ound Let M b e an

integer such that tk k M Then we can write z on the form

(k ) (k ) (k ) (k )

M 1 2

z a a a a

0 1 2

Version April

Numeration systems

Figure Base i tile with fractal b oundary

(k ) (k )

where a A for M j Let b A b e an integer so that a b for

M M

(k )

innitely many k s Let D b e the subset of those k s such that a b Let

M M

(k )

b A b e an integer so that a b for innitely many k s in D

M 1 M 1 M

M 1

and let D b e the set of those k s Rep eating this pro cess a set sequence

M 1

(k )

D such that D D and such that for all k D a b

M M M 1 j

for each j M is constructed Let k k b e an innite sequence

1 2

such that k D for j Since

j M j +1

(k ) (k )

j j

M j 1 M j M M j +1

a b b a z

M M j +1 k

M j 1 M j

b when j Since lim z z we have we get z

k k k

hz i b b b b

M 0 1 2

Example On Figure is shown the set obtained by considering com

plex numb ers having a zero integer part and a fractional part of length less than

a xed b ound in their iexpansion This set actually tiles the plane

Let C b e a nite alphab et of Gaussian integers The normalization on C is

the function

C A

c c h c i

k 0 j

j =0

As for standard representations of integers see Prop osition normalization

is a right subsequential function and in particular addition is right subsequen

tial

Proposition For any nite alphab et C of Gaussian integers the nor

malization in base n i restricted to the set C n C is a right subse

quential function

Version April

Representation of complex numb ers

Proof Let m maxfjc aj j c C a Ag and let mj j First

observe that if s Zi and c C there exist unique a A and s Zi

such that s c s a b ecause A is a complete residue system mo d

Furthermore if jsj then js j jsj jc ajj j mj j

Consider the subsequential nite transducer A over C A where

A Q E is dened as follows The set of states is Q fs Zi j jsj g

Since Zi is a discrete lattice Q is nite

E fs s j s c s ag

Observe that the edges are lettertoletter The terminal function is dened

by s hsi The transducer is subsequential b ecause A is a complete residue

system

Now let c c C and z c Setting s there is a unique

k 0 j 0

j =0

path

c a c a c a c a c a

k k k1 k1 2 2 1 1 0 0

s s s s s

k +1 k 2 1 0

k k +1

We get z a a a s and thus hz i s a a

0 1 k k +1 k +1 k 0

Representability of the complex plane

In general the question of deciding whether given a base and a set of digits

A every complex numb er is representable is dicult A sucient condition is

given by the following result

Theorem Let b e a complex numb er of mo dulus greater than and

let A b e a nite set of complex numb ers containing zero If there exists a

b ounded neighb orho o d V of zero such that V V A then every complex

numb er z has a representation of the form

z d

j m

with m in Zand digits d in A

Proof Let z b e in C There exists an integer k such that z V thus

it is enough to show that every element of V is representable Let z b e in V

A sequence z of elements of V is constructed as follows Let z z As

j j 0 0

V V A if z is in V there exist d in A and z in V such that

j j +1 j +1

z z d

j +1 j j +1

Hence the sequence z is such that

j j 0

1 j j

z d d z

1 j j

and since V is b ounded by letting j tend to innity

z d

j 0

Version April

Problems

Section

Prove that addition in the standard ary system is not left subsequen

tial

Give a right subsequential transducer realizing the multiplication by a

xed integer and a left subsequential transducer realizing the division

by a xed integer in the standard ary system

Prove the wellknown fact that a numb er is rational if and only if its

expansion in the standard ary system is eventually p erio dic

Show that any real numb er can b e represented without a sign using

a negative base where is an integer and digit alphab et

f j j g Integers have a unique integer representation Addition

of integers is a right subsequential function

Show that one can represent any real numb er without a sign using

base and digit alphab et f g Integers have a unique integer

representation Addition of integers is a right subsequential function

Generalize this result to integral bases greater than

Section

Show that the co de Y dened in the pro of of Prop osition is nite

if and only if d is nite resp is recognizable by a nite automaton

if and only if d is eventually p erio dic

If every rational numb er of has an eventually p erio dic expansion

then must b e a Pisot or a Salem numb er See Schmidt

Normalization in base See Frougny Berend and Frougny

Let s s and denote by s the real numb er s

i i1 i

Let C b e a nite alphab et of integers The canonical alphab et is A

f b cg The normalization function on C

N N

C A

is the partial function which maps an innite word s over C such that

s onto the expansion of s

A transducer is said to b e lettertoletter if the edges are lab elled by

couples of letters

Let C f cg where c is an integer Show that normalization

is a function computable by a nite lettertoletter transducer if and

only if the set

Z c fs s j s Z js j c s g

i i0 i i i

Version April

Problems

is recognizable by a nite automaton

Prove that the following conditions are equivalent

N N

i normalization C A is a function computable by a nite

lettertoletter transducer on any alphab et C of nonnegative integers

N N

A A where A f b c g is a function com ii

putable by a nite lettertoletter transducer

iii is a Pisot numb er

Section

See Hollander Let U b e a linear recurrent sequence of integers

such that lim u u for real

n n+1 n

Prove that if d is not nite nor eventually p erio dic then LU is

not recognizable by a nite automaton

If d is eventually p erio dic d t t t t set

1 N N +1 N +p

N +p

X X

N i N +pi N N +p

t X t X X B X X

i i

i=1 i=1

Similarly if d is nite d t t set

1 m

mi m

t X B X X

i=1

Note that B X is dep endent on the choice of N and p or m Any

such p olynomial is called an extended beta polynomial for Prove that

i If d is eventually p erio dic then LU is recognizable by a nite

automaton if and only if U satises an extended b eta p olynomial for

ii If d is nite then

if U satises an extended b eta p olynomial for then LU is rec

ognizable by a nite automaton

if LU is recognizable by a nite automaton then U satises a

p olynomial of the form X B X where B X is an extended

p olynomial for and m is the length of d

Section

Show that every Gaussian integer can b e uniquely represented using

base and digit set A f g if g f i i i

i i ig If each digit is written in the form

0 1 1 0

i i

1 0 0 0 1

1 1

i i i i

1 1 1 1

Version April

Problems

then for any representation the top row represents the real part and

the b ottom row is the imaginary part Every complex numb er is repre

sentable

Show that every complex numb er can b e represented using base

and the same digit set A but that the representation of a Gaussian

integer is not unique

Prove that every Gaussian integer has a unique representation of the

form d d d in base bi where b is an integer and the

k 0 1

digits d are elements of A f b g Every complex numb er is

representable See Knuth

Show that every complex numb er can b e represented using base and

2 3

digit set A f g where expi These representa

tions are called polygonal representations See Duprat Herreros and

Kla

Let b e a complex numb er of mo dulus and let A b e a nite

digit set containing Let W b e the set of fractional parts of complex

numb ers W f d j d Ag

j j

j 1

Show that W is the only compact subset of C such that W W A

Show that if the set W is a neighb orho o d of zero then every complex

numb er has a representation with digits in A

Let b e a complex numb er of mo dulus and let A b e a nite

digit set containing An innite sequence d of A is a strictly

j j 1

proper representation of a numb er z d if it is the greatest

j 1

in the lexicographic order of all the representations of z with digits in

A It is weakly proper if each nite truncation is strictly prop er Let

W f d j d Ag Show that if is a complex Pisot

j j

j 1

numb er the set of weakly prop er representations of elements of W is

recognizable by a nite automaton See Thurston Kenyon

Petronio

Representation of algebraic numb er elds See Gilb ert

Katai and Kovacs

Let b e an algebraic integer of mo dulus and let A b e a nite set

of elements of Z containing zero We say that A is an integral

numeration system for the eld Q if every element of Z has a

unique integer representation of the form d d with d in A

k 0 j

m m1

Let P X X p X p b e the minimal p olynomial

m1 0

of The norm of is N jp j Show that a complete residue

system of elements of Z mo dulo is the set f N g

Supp ose that every element of Z has a representation in A

Prove that this representation is unique if and only if A is a complete

residue system for Z mo dulo that contains zero

Supp ose that A is an integral numeration system Show that

every element of the eld Q has a representation in A

Show that A is an integral numeration system if and only if and

all its conjugates have mo duli greater than and there is no p ositive

Version April

Notes

integer q for which

q 1 q

d d mo d

q 1 0

with d in A for j q

Now supp ose that is a quadratic algebraic integer and let A

f jp j g Prove that A is an integral numeration system for

Q if and only if p and p p

0 1 0

Notes

Concerning the representation of numb ers in classical or less classical numera

tion systems there is always something to learn in Knuth Representation

in integral base with signed digits was p opularized in computer arithmetic by

Avizienis and can b e found earlier in a work of Cauchy

We have not presented here padic numeration nor the representation of

real numb ers by their continued fraction expansions see Chapter for this last

topic

The notion of b etaexpansion is due to Renyi Its prop erties were

essentially set up by Parry in particular Theorem Co ded systems

were intro duced by Blanchard and Hansel The result on the entropy

of the shift is due to Ito and Takahashi The links b etween the

expansion of and the nature of the shift are exp osed in Ito and Takahashi

and in BertrandMathis Connections with Pisot numb ers are to b e

found in Bertrand and Schmidt It is also known that normalization

in base is computable by a nite transducer on any alphab et if and only if

is a Pisot numb er see Problem If is a Salem numb er of degree

then d is eventually p erio dic see Boyd It is an op en problem for

degree Perron numb ers are intro duced in Lind There is a survey

on the relations b etween b etaexpansions and symb olic dynamics by Blanchard

In Solomyak and in Flatto Lagarias and Po onen is proved the

following prop erty if d is eventually p erio dic then the algebraic conjugates

of have mo dulus strictly less than the golden ratio Betaexpansions also

app ear in the mathematical description of quasicrystals see Gazeau

The representation of integers with resp ect to a sequence U is intro duced

in Fraenkel The fact that if LU is recognizable by a nite automaton

then the sequence U is linearly recurrent is due to Shallit We follow the

pro of of Loraud The converse problem is treated by Hollander

see Problem Canonical numeration systems asso ciated with a numb er

come from BertrandMathis Normalization in linear numeration systems

linked with Pisot numb ers is studied in Frougny Frougny and Solomyak

and with the use of congruential techniques in Bruyere and Hansel

Moreover if the sequence U has a characteristic p olynomial which is the minimal

p olynomial of a Perron numb er which is not Pisot then normalization cannot

b e computed by a nite transducer on every alphab et Frougny and Solomyak

Version April

Notes

A famous result on sets of natural numb ers recognized by nite automata is

the theorem of Cobham Let k b e an integer A set X of p ositive

integers is said to b e k recognizable if the set of k representations of numb ers of

X is recognizable by a nite automaton Two numb ers k and l are said to b e

multiplicatively independent if there exist no p ositive integers p and q such that

p q

k l Cobhams Theorem then states If X is a set of integers which is b oth

k recognizable and l recognizable in two multiplicatively indep endent bases k

and l then X is eventually p erio dic There is a multidimensional version of

Cobhams Theorem due to Semenov Original pro ofs of these two results

are dicult and several other pro ofs have b een given some of them using logic

see Michaux and Villemaire There are many works on generalizations

of Cobham and Semenov theorems see Fabre Bruyere and Hansel

Point and Bruyere Fagnot Hansel In Durand there is

a version of the Cobham theorem in terms of substitutions We give now one

result related to the concepts exp osed in Section Let U b e an increasing

sequence of integers A set X of p ositive integers is U recognizable if the set of

normal U representations of numb ers of X is recognizable by a nite automaton

Let and b e two multiplicatively indep endent Pisot numb ers and let U and

U b e two linear numeration systems whose characteristic p olynomial is the

minimal p olynomial of and resp ectively For every n if X N is U

and U recognizable then X is denable in hN i Bes When n the

result says that X is eventually p erio dic

Theorem on bases of the form n i n integer is due to Katai

and Szabo There is a more algorithmic pro of as well as results on the

sumofdigits function for base i in Grabner et al Normalization

in complex base is studied in Safer Theorem app eared in Thurston

as well as the result on complex Pisot bases presented in Problem

Representation of complex numb ers in imaginary quadratic elds is studied in

Katai We have not discussed here b etaautomatic sequences Results

on these topics can b e found in Allouche et al particularly for the case

The numeration in complex base is strongly related to fractals and tilings

Selfsimilar tilings of the plane in relation with complex Pisot bases are discussed

in Thurston Kenyon and Petronio In Gilb ert the fractal

dimension of tiles obtained in some bases such as n i is computed A general

survey has b een written by Bandt

Version April

BIBLIOGRAPHY

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