Primitive substitutive numbers are closed under rational multiplication

P

ALLAVI

K

ETKAR

L

UCA

Q. Z

AMBONI

Primitive substitutive numbers are closed under

rational multiplication

Journal de Théorie des Nombres de Bordeaux, tome

10, n

o

_{2 (1998),}

p. 315-320

<http://www.numdam.org/item?id=JTNB_1998__10_2_315_0>

© Université Bordeaux 1, 1998, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Primitive Substitutive

Numbers

are

Closed

Under Rational

_{Multiplication}

par PALLAVI KETKAR et LUCA

_Q.

ZAMBONI RÉSUMÉ. Soit

_M(r)

l’ensemble des réels 03B1 dont le

_{développement}

en base r contient une _{queue qui} est

_l’image

d’un

_point

fixe d’une substitution

_primitive

par un

_morphisme

de lettres. Nous

démontrons _que l’ensemble

_M(r)

est stable _par

_{multiplication}

_par les

_rationnels,

mais non stable par addition.

ABSTRACT. Let

_M(r)

denote the set of real numbers 03B1 whose

base-r

_digit

_expansion

is

_ultimately

_primitive

substitutive, i.e., contains a tail which is the

_image

_(under

a letter to letter

mor-phism)

of a fixed

_point

of a

_primitive

substitution. We show that

the set

_M(r)

is closed under

_{multiplication by}

rational

_numbers,

but not closed under addition.

1. INTRODUCTION

A _sequence on a finite

alphabet

A is called

q-automatic

if it is the

image

(under

a letter to letter

_morphism)

of a fixed

_point

of a substitution of

constant

_length

_q. Let

_{M (q, r)}

denote the set of real numbers whose frac-tional

_part

has a

q-automatic

base-r

digit

expansion.

J.H. Loxton and A. van der Poorten stated that each number a E

_{M(q, r)}

is either rational

or transcendental

[LoPo].

Unfortunately

a _gap has been

reported

in their

proof.

Analogously,

a _sequence on a finite

_alphabet

A is called

_primitive

substi-tutzve if it is the

_image

_(under

a letter to letter

_morphism)

of a fixed

_point

of a

primitive

substitution. Let

M(r)

denote the set of real numbers whose base-r

_digit

_expansion

is

_ultimately

_przm2tive

_{substitutive.,}

_i.e.,

contains a

tail which is

_primitive

substitutive. It is believed that a number a in

M(r)

is either rational or transcendental. This has been verified in a few

_special

cases

(see

[AlZa], [FeMa],

and

_[RiZa]).

Manuscrit reru le 15 _juin 1998.

dition. Lehr’s

_proof

relies in

_part

on a theorem of J.-P. Allouche and M.

Mend6s

Francel:

Theorem 1

_(J.-P.

_Allouche,

M. Mend6s

_France,

_[AIMe]).

Let * be an

as-sociative

_binary

_operation

on a

finite

set A and let w = be a

q-automatic

sequence in Then the induced sequence

of

partial

products

is

_q-automatic.

We will use the

following analogue

of Theorem 1

Theorem 2

_(C.

_Holton,

_L.Q.

_Zamboni,

_[HoZa]).

Let * be a

_binary

opera-tion on a

finite

set A and let w = be _an

ultimately

_primitive

substitutive _sequence in

,A.IN.

Then the induced _sequence

_of

_{partial products}

is

_{ultimately primitive}

substitutive.

2. MAIN THEOREM

Theorem 1. The set

_M(r)

is closed under

_{multiplications}

_{by Q}

but not

closed under addition.

Proof.

We

_{begin by observing}

_{that Q}

C

_M(r).

In

_fact,

the

_digit

_expansion

of a rational number is

ultimately periodic,

and a

_periodic

_sequence is

primitive

substitutive.

_{Let ~}

E

_{M(r) .}

We show that for

_positive

_integers

n and _p, both

_n6

_{and p}

are in

_M(r).

In each case we can assume that

p

0

_E

1 and

_{that fl /}

_Q.

Hence we can

write fl =

with

6k

E

_Ar

=

~0,1, ... , 1 11.

The _sequence is then

ultimately primitive

substitutive but not

_{ultimately periodic.}

We

_{begin by}

_showing

that _y =

E

_M(r).

We write _y =

with Yk

E Ar.

Then

following

[Le]

we

have

Since

for some natural number _m, and

we obtain

Consider the sequence

{(~k,r)}k 1

in the

alphabet

APr

x

_Apr.

Since the

sequence is

ultimately primitive

substitutive,

the same is true of the

sequence

{( Çk, r) }.

Let * denote the associative

binary

operation

on

APr

x

Apr

given

by2

For 1 we set

2There is a _{typographical} error in the definition of the binary operation * _{given in [Le]. It}

We next show that

_nç

E

_M(r).

Lemma 2. Let be as above. There is a

_positive

_integer

M =

M(n)

such that

_for

each k &#x3E; 0

Proof.

Fix a

_{positive integer}

1 so that _n, and set

_{(a) f}

= _{a -} for

each a E IR. Then

Note that

_{for k &#x3E;}

0 the

_quantity

l Sk

+

_{n ~°_°~ 1}

is either 0 or 1.

Let S =

(Sk

k &#x3E; 1}.

Then

rl.

For each s _e S there exist words

Vs

and

_Us

_(in

the

_alphabet

_Ar)

such that the base-r

_digit

_expansion

of

(1 -

s)

E Q

is

_{given by}

_{Since fl /}

_Q,

for each s E S there is a

positive

integer

ms so that the _sequence does not contain the subword Set

_Ms

= ₊

mslUsl

(

and M’ _E

S}.

Then for each

k&#x3E;Owehave

if and

_only

if

if and

_only

if

Thus M = I + M’ satisfies the conclusion of Lemma 2. D We return to the

_proof

of Theorem 1. Let z =

n~.

Then _{we can} write

primitive

substitutive. Let M be as in Lemma 2. Then for l~ &#x3E; 0 we have

Now since is

_ultimately

_{primitive substitutive,}

the same is true

of the _sequence

_{(~3~+1?-"

In fact if a tail of is the

image

of a fixed

_point

of a

_primitive

substitution

(,

then the

_{corresponding}

tail of

_{{(~3~+1?’’’ ?}

is the

_image

of a fixed

_point

of the

_primitive

morphism ( M+ i

defined in

_[Qu]

_(see

Lemma V. I I and Lemma V.12 in

_[Qu]).

Define § :

Ar

by

Then the _sequence

_{{~(~k,~~+1, ~ ~ ~}

_~~k+M~J~+1

is

_ultimately

primi-tive substitutive as

_required.

k+

It remains to show that

_M(r)

is not closed under addition. Let T be the

primitive

morphism

defined

_by

Let a = denote the fixed

_point

of T

_beginning

in 1 and b =

~bi}

the fixed

point

of T

beginning

in 2. Let a =

2::1 ai(10)-z

and

_/3

=

2::1

Then a

_{and Q}

are each in

M(10)

but a

+ {3 ft

M(10).

In

fact,

the

digit

3

occurs an infinite number of times in the decimal

expansion

of a

+ {3

but

not in bounded _gap. We note that for each n &#x3E; 1 the _sequence a

begins

in and b

_begins

in

_~(21)~(1).

Since

_{~T’~(12)~ _ ~T’~(21))}

it follows that for each N &#x3E; 1 we can find k =

k(N)

so that _akak+1 ... ak+N =

bk-~N ~

If c =

Icil

denotes the decimal

expansion

of a

+ (3

then

the block CkCk+1 ... consists

_only

of the

_digits

2 and 4. At the same

time,

for each n &#x3E; 1 the sequence a

begins

in

T’~(121)Tn(1)

while b

begins

in

of

_a+,~

is a minimal sequence. In

particular

is not

ultimately

primitive

substitutive. 0

REFERENCES

[AlMe] J.-P. Allouche, M. Mendès _{France, Quasicrystal ising} chain and automata _theory. J. Statist. Phys. 42 _(1986), 809-821.

[AlZa] J.-P. _{Allouche, L.Q. Zamboni, Algebraic} irrational _binary numbers cannot be _fixed

points of non-trivial constant _length or _{primitive morphisms.} J. Number Theory 69

(1998), 119-124.

[De] F.M. _Dekking, Iteration _{of maps by} an automaton. Discrete Math. 126 _(1994), 81-86.

[Du] F. Durand, A characterization _of substitutive sequences using return words. Discrete Math. 179 _(1998), 89-101.

[FeMa] S. _Ferenczi, C. _Mauduit, Transcendence of numbers with a low complexity _expansion.

J. Number _Theory 67 _(1997), 146-161.

[HoZa] C. Holton, L.Q. Zamboni, Iteration of maps by primitive substitutive sequences. (1998), to appear in Discrete Math.

[Le] S. Lehr, Sums and rational multiples of q-automatic sequences are _q-automatic.

Theo-ret. _{Comp. Sys.} 108 _(1993), 385-391.

[LoPo] J. H. _Loxton, A. van der Poorten, Arithmetic properties of automata: _{regular sequences.}

J. Reine _Angew. Math. 392 _(1988), 57-69.

[Qu] M. Queffélec, Substitution Dynamical Systems-Spectral Analysis. Lecture Notes in Math. _{1294, Springer-Verlag,} Berlin-New York, 1987.

[RiZa] R. _{Risley, L.Q. Zamboni,} A _{generalization of} Sturmian flows; combinatorial structure

and transcendence. _preprint 1998.

Pallavi KETKAR

Department of Mathematics

University of North Texas

Denton, TX 76203-5116

E-mail : _{pketkaro j} ove . acs . unt . edu

Luca Q. ZAMBONI

Department of Mathematics

University of North Texas

Denton, TX 76203-5116 E-rrtaal : luca~unt. edu