An almost-sure estimate for the mean of generalized $Q$-multiplicative functions of modulus $1$

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

J

EAN

-L

OUP

M

AUCLAIRE

An almost-sure estimate for the mean of generalized

Q-multiplicative functions of modulus 1

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{1 (2000),}

p. 1-12

<http://www.numdam.org/item?id=JTNB_2000__12_1_1_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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

1

An almost-sure estimate for the

mean

of

_generalized

_{Q-multiplicative}

functions of

modulus

1

par JEAN-LOUP MAUCLAIRE

RÉSUMÉ.

_{Soit Q =}

_(Qk)k~0,

_Q0

= 1,

Qk+1

=

qkQk, qk ~

2,

k ~ _0, une échelle de

Cantor,

ZQ

le groupe _compact

03A00~j Z/qjZ,

et 03BC sa mesure de Haar normalisée. A un élement x

_{of ZQ}

écrit x =

{a0,a1,a2...},0 ~ ak ~

_{qk+1-1,k ~}

_0,

on associe la suite

xk =

03A3o~j~k ajQj .

On montre que

si g

est une fonction

Q-multiplicative

unimodulaire,

alors

lim xk~x(1/xk

_03A3n~xk-1

g(n) -

03A0 0~j~k 1/qj

03A30~a~qj-1

g(aQj) = o

03BC-p.s.

ABSTRACT Let

_{Q =}

_(Qk)k~0,

_Q0

= _1,

Qk+1= qkQk, qk ~

2,

be a Cantor

scale,

_ZQ

the

compact

_projective

limit group of the

groups

Z/QkZ,

identified to

03A00~j~k-1

Z/qjZ,

and let 03BC be

its

normalized Haar measure. To an element x =

{a0

_a1,

a2, - - -},

0 ~ ak ~ qk+1 - 1, of

_ZQ

we associate the _sequence of

_integral

valued random variables xk =

03A30~j~k

ajQj.

The main result of

this article is

_{that, given}

a

complex Q-multiplicative

_{function g}

of

modulus 1, we have

lim xk~x

(1/xk

_03A3n~xk-1

_g(n)-

03A00~j~k 1/qj

_{03A3 0~a~qj}

_g(aQj))

= 0 03BC-a.e.

1. INTRODUCTION

Let N be the set of

_{non-negative integers,}

and let

_Q

=

Qo

=

1,

be an

_increasing

_sequence of

_positive

_{integers. Using}

the

greedy algorithm,

to _every element n of

_N,

one can associate a

_{representation}

which is

_unique

if for _every

_K,

The

_simplest

_examples

are the

_{q-adic scale, q integer,}

_{q &#x3E; 2,}

and its

gen-eralization,

the Cantor scale

_Qx+1

=

Qo

=

1,

2,

k &#x3E; 0. In this

article,

we are concerned with the Cantor scale. For a

given

integer n &#x3E; 1,

we denote

by k(n)

the maximal index k for which is different from

zero. The

integers ek (n)

are the

_digits

from n in the basis

Q.

We recall

that if G is an abelian _group, a G-valued arithmetical function

f

such that

is called a

_{Q-additive function,}

an extension of the notion of

_q-additive

function introduced

_by

A. O. Gelfond in the

_q-adic

case

~4~.

We recall that a real-valued _sequence

f (n)

has an

asymptotic

distribution if there exists

a distribution function F such that for all

_{continuity points}

x of

F,

the

probability

measures defined

by

I-£N(X)

=

N;

f (n)

x}

tends

to as N tends to

_infinity.

In the case of the

_q-adic

_scale,

_necessary

and sufficient conditions for the existence of an

_asymptotic

distribution for

a real-valued

_q-additive

function have been

_given

_by

H.

_Delange

in 1972

[3].

J.

_Coquet

_[2]

considered in 1975 the same kind of

problem

in cases of

Cantor scales and obtained

_mainly

sufficient conditions. In both _cases, it

appears essential to have information on the difference

where

_g(.)

is _any

_{Q-multiplicative}

function of modulus

_1,

and more

pre-cisely,

to

_get

a characterization of

In

_fact,

if the _sequence is

_bounded,

the relation 1 is

_always

true.

But if is

_unbounded,

the situation is

_quite

different. In

_(1~,

G. Barat

constructs a

Q-multiplicative

function h with values 1 or -1 such that

is less than or

_equal

to zero. This difference is due to the existence of a

first digit phenomenon,

unavoidable for unbounded _sequence as

remarked

_by

E. Manstaviëius in a recent article

_[6].

Let

_ZQ

denote the _group of

_{Q-adic integers,}

considered as the

_compact

projective

limit _group of

_Z/QkZ

and identified to

_{Z~qkZ (see}

_[5],

_p.

109).

The

_products

-

-are

_clearly

related to _{this group in the}

_following

_way: an element a of

ZQ

can be written a =

(ao, al, ... ), 0

_{qk -}

1, 0

k,

and we _may

identify

an element of N with an element of

_ZQ

which has

only

a finite

number of

_digits

different from zero. For all a =

(ao, al, ... )

belonging

to

_ZQ,

we define on

_ZQ

the _sequence of N-valued random variables

xk(-)

given by

ajQj,

the

_compact

group

_ZQ

being

endowed with its normalized Haar measure _M, and

clearly

In this

_article,

we show

_{roughly speaking}

that

_although

the relation 1 is not

always

true

_according

to the

_example

of G. Barat

_(for

unbounded sequence it is almost

_surely

true for a

_path

chosen at random.

2. RESULTS 2.1. Main theorem.

Theorem 1.

_{Let 9}

be a unimodular

Q -multiplicative

function

and set

Then,

the relation

holds _{p,-a. e.}

2.2.

_Consequence

of Theorem 1.

Theorem 2. Let G be a metrizable

locally

_compact

abelian _group with

group law denoted

by

+. r denotes the dual _group

_of

G endowed with its Haar measure _m, and let

_f(n)

be a G-vadued

Q-additive function.

Given a _sequence

A(k)

in

_G,

we denote

by

Ft

the distribution

of

the G-valued

function defined

on

ZQ

by

t ~

( f(zk(t)) -

_A(k)),

and

_by

the measure

consisting in

a unit mass at the

_point

a.

The

_following

assertions are

equivalent:

i)

there exists a _sequence

A(k)

in G and a

_probability

measure v on G

such that the _sequence of distributions

_FA

_converges

_vaguely

to v

_(i.e.,

limk fG pdf/

=

fG

cpdv

for all continuous _{maps cp :} G - C with

_compact

support;

ii)

there exists a _sequence

A(k)

in G and a

probability

measure v on G

such that _~-a.e., the _sequence ofrandom measures

2 ~.~

converges

vaguely

to v as k tends to

_infinity;

iii)

the set X of characters

_{g of r for}

which there exists an

_integer

N(g)

such that

_{Mj(g) 0}

0 is not

_{rn-negligible.}

Remarks.

₁₎

Assertion

_iii)

is

_always

satisfied if G is a

_compact

metrizable

group, for X is not

empty

(it

contains the trivial

_character),

and

conse-quently

is not

_{m-negligible.}

2) Necessary

and sufficient conditions for the

_continuity

of v can be

easily

found,

since v appears as a convolution of measures on

_ZQ:

in

_fact,

the same

method as in

[7]

(p. 84-87), gives

that X is a closed and _open

_subgroup.

Denoting by H

the

_orthogonal

of X and

_by

_TH

the canonical

_projection

G ~

_G/H,

the measure v is not continuous if and

_only

if H is finite and

2.3. Proof of Theorem 2. A

_{straightforward}

_adaptation

of the

_argument

given

in

_[7]

_{(p 84-87)}

leads

to,

primo

if one of the

_assumptions

i), ii), iii),

holds, then,

X is a closed and _open

_{subgroup of r;}

and

_secundo,

there exists a

probability

measure v on G and a G-valued _sequence

(A(k))k

such that

for

_{all g}

in

_r,

the _sequence

tends to

_v(g)

where V is the Fourier transform of v. This is due to the fact that

_{for g}

in X there exists an

N(g)

for which the relation

11

0,

j&#x3E;N(g) holds. Hence we

_get

by

Theorem 1 that for all _g, the _sequence

converges to

v(g)

~-a.e.

Next,

we use the Fubini theorem on the measured

space

(r

x

_{ZQ ,}

in an essential _way,

_by

_saying

that since r is countable

and so, p,-a.e., the

A(k))}k

converges to

v(g)

m-a.e.. In order to _prove that _p,-a.e., the _sequence

converges

vaguely

to v, it suffices to show that for _any real-valued

contin-uous function F defined on G whose

_support

is

_compact,

the _sequence

converges to

_v(F).

This can be done as follows. Take _{any c} &#x3E;

_0;

_by

assumption

on

F,

there exists

V,

a

_{symmetric neighborhood}

of the

_origin

in

_G,

such that for all t in G and all u in

_V,

one has

IF (t

+

_{u) -}

_F(t)1

_I

e.

Denoting by

M the Haar measure on G normalized with

_respect

_{to m,} we

have

The function

_{Fv (t)}

defined

_by

is the convolution

_product

of F with the characteristic function of V

nor-malized

_by

the constant

_Therefore,

the Fourier transform

FV

is

By

the

_Lebesgue

dominated _convergence

_theorem,

Now,

since v is a

probability

measure the _sequence

is bounded in modulus

_{by 2E;}

this

_implies

Therefore,

the sequence

a.e. to v.

converges

vaguely

it-k

3. PROOF OF THEOREM 1 Notation and conventions

Given an

arbitrary

arithmetical function

f ,

we set

Notice that we have the

_identity

multiplicative

function

_{f ,}

By convention,

the result of a summation

(resp.

a

product)

on an

_empty

set will be 0

_(resp.l).

A - Toolbox.

Proposition

1.

_{Let g}

be a

Q -multiplicative function of

modulus 1 and

sequences

of complex

nuTnbers

of

modulus 1 such that

Proof.

By

our

assumption,

all the

complex

numbers are different

from zero. Put ai = where

~(’)

is the

complex

con-jugate of g(.).

The

_product

is

_equal

to

_lmj(g)l

_]

and the sequence

IlMk+lll.k

is

_convergent.

_Therefore,

From

we

_get

a fortiori that the

series

]

verges and since

ig(aQk).ak I =

1,

we deduce

According

to

_{Proposition 1,}

we introduce the _sequence of arithmetical

func-tions

_gk(n)

defined

_by

where n is written in base

Q

as n =

L;=o

ajQko

This means that if

k(n)

is

the index of the last

_digit

of n which is different from _zero, is

equal

to

We extend

_gk

_by

_gk(x)

=

9*

0 _Xk which _we also denote

_gk.

_Moreover,

for

simplification,

we shall use the notation

g*(aQj)

=

Proposition

2.

_If

the _sequence does not _converge to

_0,

there exists a subset

Eoo

_{of ZQ}

such that

1L(Eo,,)

= 1 and

for

every a =

Proof.

The _sequence of finite _groups

_Z/QkZ,

k &#x3E;

_0,

induces a filtration

on the

it-measured

_space

_ZQ,

and the

complex-valued

_sequence of

adapted

functions for this filtration defined

_by

is a

martingale.

Since we have

and

is

_bounded,

this

_martingale

is bounded and _{so, it converges p,-a.e.} But the

sequence

is

_convergent.

Hence we obtain that the _{sequence converges &#x3E;}

-a.e. D

Proposition

3.

_{If the}

_sequence does not tend to

_0,

there exists

a subset

F 00

of

ZQ

such that = 1 and

for

every x =

(ao(x),

_al

(x), ...)

in

_F~,

one has

Proof.

Assume that the _sequence does not tend to 0.

_Using

the same notations as in

_Proposition

_2,

we have

_{by Proposition}

1

Let ak be defined

by

For a~ in

_ZQ,

we write

x =

(ao(x), al (x), ...), 0

-:5

_{qk -}

_{1, 0}

k and we remark

_that,

on

the _sequence of the

_ak(x)

different from

_0,

one has

Since

_Ek

Uk +oo, it is known that there exists an

increasing

posi-tive function h

_tending

to

_infinity

when k tends to

_infinity

such that We consider the set

_F(h)

nn.-u

of

_points

x in

_ZQ

such that for all

k,

the

_inequality

holds,

where

_{( ~ ~ ] denotes}

the

_integral

_part

function. This set

_F(h)

is

_closed,

and its measure

p,(F(h))

is

equal

to

and we have

Now,

we remark that this last

product

can be written

FC2013U

we consider the condition

is not

_0,

then we have

and in this case, we

_get

., .-

, ,, .. , , , ,

, The case where = 0

remains. We have 0

_qkQkjt(k)

_1,

i.e. _qkQk

_llh(k).

Hence

To obtain our

result,

we remark that the _sequence of functions

hr

indexed

by positive integers

r and defined

by

hr(n) =

h(n)

if n &#x3E; r and

h(n)r-1

otherwise,

satisfies the same

_requirements

as h.

_Now,

the _sequence of closed

sets

_F(hr)

is

_increasing

with r and lim 1. This

_gives

imme-r-t+oo

diately

that

_{F ,}

the union of the is a measurable set of measure 1.

Now,

if x

belongs

to it

_belongs

to some

F(hr)

and as a _consequence,

along

the

_{sequence k}

such that

_0,

we have

Proposition

4.

_If

the _{sequence converges} to zero, then

- ,..

Proof.

This

_Proposition

is due to

_J.Coquet

_2.

D B- End of the

_proof

1- First case: the _sequence

{Mk+1 (f )}x

tends to zero.

From

_Proposition

_4,

lim

_{E f (n)}

=

0,

and tends to

_infinity

N +-

&#x3E;-a.e. due to the fact that

_xk(a)

is bounded if and

_only

if a has

only

a

finite number of nonzero

_digits.

This means

_exactly

that a is an

integer;

but = 0.

2- Second case: the _sequence

{Mk+1(f)}k

does not tend to zero.

We consider the intersection of the sets

_Eoo

and

_{F 00}

_given

in

_Proposition

2 and

_Proposition

3

_{respectively.}

Notice that = 1. Our aim is

to prove that for

_{every ~}

in

_Eoo

fl

F 00

The _sequence of functions k H

_{9Z (n)}

and the constants _aj are defined as

in

_Proposition

2.

_{Let ~}

be an element of

_Eoo

f1 and denote

_Xk(Ç)

_by

xk for short. We have:

and

_by

iteration

when j

tends to

_{infinity. Since g}

is of modulus 1 and

I is bounded

_by

_1,

Consequently

Since ~ belongs

to

_E~,

_converges, and as a _consequence, the

se-quence tends to 0 when k and

.7, ~ ~ ~

tend to

infinity independently.

This

_implies

and _so, when k - _+oo,

Finally

It then follows that

To obtain the

_result,

it is

_enough

to notice that from

1 and

replacing

and this leads to , I

REFERENCES

[1] G. _Barat, Echelles de numération et fonctions arithmétiques associées. Thèse de doctorat,

Université de Provence, Marseille, 1995.

[2] J. Coquet, Sur les fonctions S-multiplicatives et S-additives. Thèse de doctorat de

Troisième Cycle, Université _{Paris-Sud, Orsay,} 1975.

[3] H. Delange, Sur les _{fonctions q-additives} ou _{q-multiplicatives.} Acta Arithmetica 21 (1972),

285-298.

[4] A.O. Gelfond, Sur les nombres _qui ont des _propriétés additives ou _{multiplicatives} données.

Acta Arithmetica 13 _(1968), 259-265.

[5] E. Hewit, K.A. Ross, Abstract harmonic _{analysis. Springer-Verlag,} 1963.

[6] E. Manstavicius, Probabilistic theory of additive functions related to _systems of

numera-tion. New trends in _Probability and Statistics Vol. 4 _(1997), VSP BV &#x26; TEV, 412-429.

[7] J.-L. _Mauclaire, Sur la _repartition des fonctions q-additives. J. Théorie des Nombres de

Bordeaux 5 _(1993), 79-91.

Jean-Loup MAUCLAIRE

Th6orie des Nombres _{(UMR 7586)}

15 rue du Chevaleret

F-75013 Paris cedex