The distribution of the sum-of-digits function

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

M

ICHAEL

D

RMOTA

J

OHANNES

G

AJDOSIK

The distribution of the sum-of-digits function

Journal de Théorie des Nombres de Bordeaux, tome

10, n

o

_{1 (1998),}

p. 17-32

<http://www.numdam.org/item?id=JTNB_1998__10_1_17_0>

© Université Bordeaux 1, 1998, tous droits réservés.

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

The Distribution of the

_{Sum-of-Digits}

Function

par MICHAEL DRMOTA et JOHANNES GAJDOSIK

RESUME. Dans cet

_article,

nous démontrons que la fonction

"somme de chiffres"

_(relative

à des recurrences linéaires finies et

infinies

_{paxticulieres)}

satisfait à un theoreme central limite. Nous

obtenons aussi un théorème limite local.

ABSTRACT.

By using

a

_generating

function

_approach

it is shown that the

_{sum-of-digits}

function

_(related

to

_specific

finite and

in-finite linear

_recurrences)

satisfies a central limit theorem.

Addi-tionally

a local limit theorem is derived.

1. INTRODUCTION

Let G = be _a

strictly increasing

sequence of

integers

with

_Go

= 1.

Then _every

_{non-negative integer n}

has a

(unique)

_proper

_{G-ary digital}

expansion

with

_{integer digits}

0

_provided

that

for 0. The

_{sum-of-digits}

functions

_sG(n)

is

_{given by}

and the aim of this _paper is to

_get

an

insight

to the distribution of

sG ~n),

i.e. to the behaviour of the numbers

Version finale _reçue le 20 janvier 1998.

Key words and _{phrases. digital expansions,} central limit theorem.

It is also _very convenient to consider a related _sequence of

(discrete)

random

variables

_XN,

N &#x3E;

_0,

defined

_by

Expected

value and variance of

_{X N}

are

_{given by}

There is a vast literature

_{concerning asymptotic properties}

of

_{EXN and}

VXN

and on the distribution of

XN.

Asymptotic

and exact formulas for

_EXN

are due to Bush

_[3],

Bellman and

_Shapiro

_[2],

_Delange

_[5],

and

_Trollope

_[20]

for q-ary

digital

expan-sions and due to Peth6 and

_Tichy

_[17]

and Grabner and

_Tichy

_{[12, 13]}

for

G-ary

digital

expansions

with

_respect

to linear recurrences.

Correspond-ing

formulas for

_higher

moments and for the variance

_VXN

can be

found in

_Coquet

_[4],

Kirschenhofer

_[15],

_Kennedy

and

_Cooper

_[14],

Grab-ner,

Kirschenhofer, Prodinger,

and

Tichy

(11~,

and in Dumont and Thomas

(7J .

The

_asymptotic

distribution of

_{X N}

and related

_problems

are discussed

in Schmidt

_[19],

Schmid

_[18],

_Bassily

and Katai

_[1],

and in Dumont and Thomas

_[8].

The main _{purpose of this paper is} to _prove

_{asymptotic normality}

_(of

the distribution of

_XN)

by

the use of

generating functions,

where it is also

possible

to derive a local limit law.

(A

similar

approach

was used in

[6].)

2. RESULTS

In the

_present

_paper we will deal with basis _sequences G = which

satisfy specific

finite or infinite linear recurrences.

2.1. Finite Recurrences. In the first case we will make the

following

assumptions:

1. There exist

_{non-negative integers}

_{ai, 1 i r,} such that

_{(for j &#x3E; r)}

2.

In section 3 we will show that the above

assumptions

_imply

that the

char-acterictic

_polynomial

has a

_unique

root a of maximal modulus which is real and

positive

(i.e.

all

other roots

a’

of

_P(u)

_satisfy

_a)

and that

for some constant C &#x3E; 0.

.Remark.

_Usually

_(e.g.

see

[12])

it is assumed that

_{a2 &#x3E; ~ ~ ~ &#x3E;}

_ar &#x3E; 0

j

and that

_{G j}

+ 1

_{for j}

r. In this case all

_assumptions

are

i=l

satisfied.

_{(Furthermore, P(u)}

is irreducible and a is a Pisot

number.)

If the _sequence of

_ai,1

i =5 r, is not

decreasing

then the situation is more

complicated,

e.g. if al = ar =1

and ai

= 0 for i

~

_{l, r}

then condition 3. is

satisfied,

too.

_However,

if r

= 4,

al = _a3 = _a4 =

1,

and _a2 = 0 then 3. is

violated.

2.2. Infinite Recurrences. In the second case our

_{starting point}

is

Parry’s

a-expansion

of 1

_(see

_[16])

where a &#x3E; 1 is a real number

and ai,

i &#x3E; 1 are

_positive

_integers.

(In

the case of

ambiguity

we take the infinite

representation

of

1.)

The _sequence

G = is now defined

_by

If we further set

and

then

and it follows that zo =

1/a

_&#x3E; 0 is the

only singularity

on the circle of

convergence

izi

=

zo, which is a

simple pole.

Hence

2.3.

_{Asymptotic Properties.}

First we state a theorem

_concerning

ex-pected

value and variance

_{of X N}

_(defined)

in

_{( 1.1 ) .}

_Actually

this statement

is more or less a collection of well known facts

(see

[5,

_17,

_{13, 15, 7,}

10~).

More

_precisely,

much more is known about the

_following

_{0 ( 1 )-terms.}

There-fore we will not

_present

a

proof.

Theorem 2.1.

_Suppose

that G =

satisfies

_a

finite

_or

infinite

linear

recurrence

of

the above

_types.

Set

and let

_1/a(z)

denote the

_(analytic)

solution u =

1/a(z)

of

the

_equation

for

z in a

_sufficiently

small

(complex)

_{neighbourhood of}

_zo = 1 such that

a(l)

= _a. Then

and

where

and

Our main result concerns the distribution

_properties

of

_XN.

We _prove

asymptotic normality

in the weak sense and

_provide

a local limit law.

Theorem 2.2.

_Suppose

that G =

satisfies

_a

finite

_or

infinite

linear

recurrence

of

the above

_types.

If

a254

0 then

for

_{every 6} &#x3E; 0

uniformly for

all real x as N -~ oo.

Furthermore

In section 3. we collect some

preliminaries

which will be used in sections 4.

and 5. for the

_proofs

of

_(2.2)

and

_(2.3).

3. PRELIMINARIES

3.1. Finite Recurrences. We will first collect some basis facts which will

be needed in the

_sequel.

Lemma 3.1.

_Suppose

that

_{Go, G1,... ,}

_Gr-1

are

_positive,

that

_{Gj =}

aigj-i

for j &#x3E;

r, where

0,

1

i r, and that 1 :

r

ai i4

0}

=1. Then the characterictic

_polynomial

_P(u)

= ur - has

i=l

a

_’Unique

root

_of

maximale modulus which is real and &#x3E; 1.

_Furthermore,

I

for

a real constants C &#x3E; 0 and some 6 &#x3E; 0.

Proo f .

First we show that

P(u)

has a

_{unique positive}

real root a &#x3E; 1 of

r

maximal modulus. Set

_{G(u) =}

1-

_ajuj.

Then

_{G (u)}

is

j=l

strictly increasing

for real u &#x3E; 0. Since

_G(o)

= 0 and lim

G(u)

= oo there

’

uniquely

exists uo &#x3E; 0 with

_G(uo)

= 1. Since

Gn

is

strictly increasing

_we

have and

_consequently

_uo 1.

_Furthermore,

_{G’ (uo )}

=

Thus,

a =

1/uo

_&#x3E; 1 is a

_simple

root of

_P(u).

then

Furthermore,

if

_lul

= _{uo uo} then the

gcd-condition

1 :

ai 0

0}

= 1

_implies

that

Consequently,

there are no roots of

_P(u)

other than a with modulus &#x3E; a.

Next it is clear that

_Gj

has a

representation

of the form

(3.1)

for some real C. We

_only

have to show that C &#x3E; 0. For this _purpose define

for j &#x3E;

r. Then is multilinear and monotonic in all

Hence, by setting

t we obtain

ThusC&#x3E;0. D

Lemma 3.2.

_Suppose

that G =

(Gj)j&#x3E;o

satisfies

the above conditions 1. -~.

has

_digits

Suppose

that n has the

_digital

_expansion

has

_digits

Proof.

Since

_kgj-i

+ m

aigi

we obtain for 1 z’ i

_by

condition 3.

Thus,

El (n)

=

aj-1

for j -

i I

j.

Similarly,

implies

= k and

consequently

=

El (rn)

for 0 t

_{j -}

i. This

completes

the

_proof

of the first

_part.

which

_gives

=

El (n)

for j

1 L.

Finally

provides

fj(n’)

= k and =

ei (m)

for 0 1

_j.

Next let

and set

and

Lemma 3.3.

_{For j &#x3E;}

r we have

Proof.

First observe that the set

_{~

e Z :

_{0 n}

_{( j &#x3E;}

_r)

can be

represented

as a

disjoint

union of the form

Thus

_by

Lemma 3.2

which

_provides

_(3.2).

Corollary.

We have

3.2. Infinite Recurrences. In the case of

digital

_expansions

which are

related to

_Paxry’s

_a-expansion

we have similar

_properties.

Lemma 3.4.

_([13])

Let G =

given by

(2.1).

Then _a

finite

se-quence

(,Eo, Cl 7...

of nonnegative integers

constitute the

G-ary digits

I

if

and

_only

_if

for

k =

_{0, l, ... , L,}

where " " denotes the

_{lexicographic}

order.

Remark. Note that

_(3.3)

_implies

that Lemma 3.2 holds for the infinite _case,

too.

As above let

By using

(3.3)

(or

the

_{corresponding}

version of Lemma 3.2 for the infi-nite

_case)

we obtain a similar

_{representation}

of

B(z, u)

as in

_Corollary

of

Lemma 3.3 in the case of finite recurrences.

Lemma 3.5.

_([13])

We have

in which

_G(z,

_u)

is

_defined

in Theorem 2.1 and

3.3.

_Composition.

A similar

_procedure

which led us to recurrences for

bj,k (resp.

to

_generating

function identities in

_Corollary

of Lemma 3.3 and

to Lemma

_3.5)

can also be used _{to extract aNk} from =

aG; k.

Lemma 3.6.

_Suppose

that

is the

_{G-ary digital}

_expansion

_of

_N,

_{where j}

Then

Proof. By

Lemma 3.2

_(which

is also valid in the case of infinite

recurrences)

we have

Thus

_(3.4)

follows. 0

4. GLOBAL LIMIT LAW The first

_step

is to obtain _proper information of

Proposition

4.1.

_Suppose

that G =

satisfies

a

finite

or

infinite

recurrence

_of

the above

_types.

Then

unif ormly for z

contained in a

_sufficiently

small

_{complex neighbourhood of}

zo = 1

as j

_{--~ oo,} where

a(z)

is

defined

in Theorem 2.1 and

C(z)

is an

analytic function

with

_C(l)

= C

resp.

C(1)

=

C’.

Proof. Firstly,

let

_Gn

_satisfy

a finite linear recurrence of the above

_type.

Then 1- =

urP(u"1),

where

P(u)

is the characteristic

polynomial.

Thus = 0.

Furthermore,

since 0 for real and

non-negative

z, u there exists an

_analytic

function

(for z

in a

_sufficiently

small

_complex

_{neighbourhood}

_{of zo}

_{= 1)}

with

_G(z,

_1/a(z))

= 0 and

a(1) _

a.

_Similarly

we have

G(l, a-’) =

0 and

’9 G(z,

u)

0 in the case of

infinite linear recurrence.

Thus,

there also exists an

_analytic

function

a(z)

with = 0 and = _a.

In both cases there exist

_analytic

functions

such that

for _{z, u in} a

_complex

_{neighbourhood}

of _zo =

1, uo

= a. Since

1/a

=

1/a(1)

is the

_unique

_{(and simple)}

zero of

G(l, u)

= 0 on the circle

_lul

=

1/a

and since there are no zeroes for

Jul

1/a

the function

G1(z,u)

can be

analytically

continued to

_Jul

_(for

some

sufficiently

small e &#x3E;

0)

if

generality

we _may assume that

11/a(z)1 ~ 1/a + E/4.

Hence,

R1(z,u)

can

be

_analytically

continued to the same

_region

and we obtain for

_Jul

_1/a+e

where

_{C(z) =}

_{-a(z)P(z,1/a(z))/G1(z,1/a(z)).}

_{Finally, by Cauchy’s}

for-mula we

_get

with some 9 &#x3E; 0. This

_completes

the

_proof

of

_Proposition

4.1. CJ With

_help

of

_Proposition

4.1 and Lemma 3.6 we can _prove

_asymptotic

normality

Observe that

is the characteristic function of

XN.

Proposition

4.2.

_Suppose

that

_{a2 54}

0 and _{set I-ZN} =

EXN

and

0,2

=

VXN.

Then

_for

_every E &#x3E; 0 we have

_{uniformly for}

(logN)1/2-ë

Proof.

Set

_{f (z)}

=

loga(ez)

in an _open

_{neighbourhood}

of z = 0. Then _we

have

with

rem

_2.2).

_Hence,

_{by using}

_Proposition

4.1

in an _open

_{neighbourhood}

of t = 0 in R. Now _suppose that

.,-v

by

Lemma 3.6 and the trivial estimate we have

Now observe that

and that

Hence

Let c &#x3E; 0 be a

(small)

real number and let x be defined

by

j",.

Then ~

_j-’

and

_consequently

Since

_jo

=

(log N)/(log a)

₊

0(1)

this

implies

(4.2)

directly

(log

N)ë/3.

Furthermore,

since

for

_I

_(log

N)1/2-ê

and a

sufficiently

small c &#x3E; 0 we

finally

obtain the full version of

_(4.2).

0

Proof.

Set

Then

_by

Esseen’s

_[9,

p.

32]

inequality

we have

Choosing

T =

(log

N)1/2-E

_we

directly

obtain from

Proposition

4.2 and

by

applying

the estimate

for

_(log

_N)-2

that

Hence,

(2.2)

follows. 0

5. LOCAL LIMIT LAW

In order to _prove a local limit law for

_XN,

i.e. the second

_part

or Theo-rem

2.2,

we need more

precise

information on the behaviour of

bj(z).

Proposition

5.1.

_Suppose

that G =

satisfies

_a

finite

or

_infinite

recurrence

of

the above

_types.

Then there exist ₁₇ &#x3E; 0 and J &#x3E; 0 such that

uniformly for

It I

q, where

C(z)

and

a(z)

are as in

Proposition

l~.l,

and

uniformly

forq

Proof.

Obviously, (5.1)

follows from

_(4.1)

_{for some r} &#x3E; 0.

For the

_proof

of

_(5.2)

_{we just}

have to observe that

_{u)1 I}

1 if

_H

_{1, z 54}

_1,

and

_Jul

_1/a.

_Hence,

_by

_continuity

there exist 6 &#x3E; 0 and T &#x3E; 0 such that

_{~1 - G(u,}

_I

&#x3E; T

_uniformly

for

_(real)

t

_{with 17}

_~t~

_I

7r and

_{(complex) u}

with

_Thus,

is

_analytic

_{(and therefore}

bounded)

in this range and we obtain

with some 6 &#x3E; 0. C7

With

_help

of

_Proposition

5.1 it is

_possible

to derive

_asymptotic

expan-sions for via saddle

_{point approximations.}

Proposition

5.2. We have

uniformly f or

all

_j,

k &#x3E; 0.

Proof.

We

_again

use

Cauchy’s

formula

Since

we

just

have to evaluate

it follows that there

Finally,

This

_completes

the

_proof

of

_Proposition

5.2. 0

Finally Proposition

5.2 and Lemma 3.6 can be used to

_complete

the

_proof

of Theorem 2.2.

L-1

Proof.

As in the

_proof

of

_Proposition

4.2 we _suppose that N

= E

_{el Gj,}

1=0

(with

jo

&#x3E;

_j1

&#x3E; ... &#x3E;

_{jL-1 and el}

&#x3E;

₀₎

is the

_{G-ary expansion}

of N.

Furthermore,

let e &#x3E; 0 be a

(small)

real number and let x be defined

by

where we have used _{p N} = ₊

O(1)

and

a 2

=

jou2

₊

O(1).

_Hence,

from

we obtain

If

jð/2logjo

then we have for 1 x

which

_finally

_gives

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