Digital expansion of exponential sequences

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

M

ICHAEL

F

UCHS

Digital expansion of exponential sequences

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{2 (2002),}

p. 477-487

<http://www.numdam.org/item?id=JTNB_2002__14_2_477_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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

Digital expansion

of

_exponential

_séquences

par MICHAEL FUCHS

RÉSUMÉ. On s’intéresse au

développement

en base _q des N

pre-miers termes de la suite

_{exponentielle}

an. En utilisant un résultat

dû à Kiss et

_Tichy,

nous montrons _que le nombre _moyen

d’occur-rences d’un bloc de chiffres donné est

_égal

_{asymptotiquement}

à

sa valeur

supposée.

Sous une

_hypothèse

_plus

forte nous montrons

un résultat similaire en ne considérant seulement les

(log N)

3/2-~,

avec ~ &#x3E; _0,

_premiers

termes de la suite an.

ABSTRACT. We consider the q-axy

_digital

expansion

of the first N terms of an

_exponential

_{sequence an.}

_Using

a result due to Kiss and

_Tichy

_[8],

we _prove that the average number of occurrences

of an

_arbitrary

_digital

block in the last

_{c log N digits}

is

asymptot-ically equal

to the

_expected

value. Under _stronger

_assumptions

we _get a similar result for the first

(log N)3/2-~

_digits,

where ~ is

a

_positive

constant. In both

_methods,

we use estimations of

ex-ponential

sums and the _concept of

_discrepancy

of real _sequences

modulo 1

_plays

an

_important

role.

1. Introduction

In this _paper, we write

_{N, Z,}

R for the sets of

_positive

_{integers, integers,}

and real numbers. With

_P,

we denote the set of

_primes

and for an element

of P we

usually

write _p. For a real number _x, we use the standard notations

e(x) =

~x~

for the fractional

_part

_{of x,}

_{and Ilxll}

for the distance from

x to the nearest

integer.

Let q &#x3E;

2 be an

_integer.

We consider for n E N the _q-ary

_{digital expansion}

We are

_going

to introduce further

_notations,

which we use

_throughout

this

paper. We start with

which is the number of

_{changes of digits (or}

the number

_{of blocks)}

in the

dig-ital

_expansion

of n.

_Furthermore,

we write for

_arbitrary

_digits

_{eo, ei, " - , es}

Manuscrit _reçu le 12 octobre 2000.

with s &#x3E;

_{0, 0 e~ ?20131,0~~5,}

not all

_{digits equal}

to 0 and

_integers

for the number of occurrences of the

_digital

block _eo in the

_digital

expansion

of n between a and b.

(If

a s then we start with i = _s and if

we omit a and b then we assume i &#x3E;

_s.)

If we use the word

_digital

_block,

we

_always

assume that at least one

digit

is not

_equal

to zero.

Finally,

we use the well known notation of

for the

_{sum-of-digits}

function.

In this _paper, we consider the _q-ary

_expansion

of an

_exponential

_sequence

an,

where a &#x3E; 2 is an

_integer.

In a recent work

_{Blecksmith, Filaseta,}

and Nicol

_[5]

_proved

the

_following

result:

Later

_Barat,

_Tichy,

and

_Tijdeman

_[3]

_gave a

quantitative

version of the

above

_result,

_by

_applying

Baker’s theorem on linear forms in

_logarithm

(see

for instance

_[1]

or

_[2]).

_They

_proved

the

_following

result:

Theorem 1. Let a and _q be

_integers

both &#x3E; 2. Assume that

_loga

_{q is}

irrational. Then there exist

_{eflectively computable}

constants _CO and _no, where _co is a

_positive

real number and _no is an

_integer.,

such that

for

7~0’

Clearly,

as a _consequence of this

_result,

we obtain the same lower bound

for the

_{sum-of-digits}

function

_S,(an)

and for the mean value of the

sum-of-digits

function of an

_exponential

_sequence.

Corollary

1. Let _{q, a} be as in Theorem 1. Then we

_have,

as ~V 2013~ OOY

One aim of this paper is to

_improve

this lower bound. More

_generally

we are interested in the behaviour of the

_following

mean value

where _eo is an

_arbitrary

_digital

block. Of _course, results about the behaviour of

₍₅₎

_imply

results about other

_interesting

mean

values,

_e.g., the

mean value of the

sum-of-digits

function and the mean value of the number

of

_changes

of

_digits.

First,

we consider

only

the last

digits

in the

digital expansion

of the

exponential

sequence.

By using

a result due to Kiss and

_Tichy

_[8],

we can

prove that the average number of occurrences of an

_{arbitrary digital}

block

is,

except

of a bounded error

_term,

asymptotically equal

to the

_expected

value. In detail the

_following

theorem holds:

Theorem 2. Let _{a, q} be

_integer

both &#x3E; 2 such that

_log,

_{q is}

irrational. We consider a

digital

block _{es es _ 1 - - - eo} with s &#x3E;

0, 0

ei

:5

q -

1, 0

i s .

There exists a

positive

real constant _~, such that we

_have,

as N ---~ _oo,

with

As an _{easy consequence,} we can remove the

_{log log}

N factor in the lower bound of

_Corollary

1.

Corollary

2. Let a, q and eses-i ... eo be as in Theorem 2. Then te

_have,

as ~V 2013~ _oo,

and

_consequently

and

Next,

we consider the first

digits.

Here it seems to be more convenient

to use the

_{stronger assumption}

(a, q) =

_1,

instead of

_{loga q}

E

_Then,

we are able to _prove a result similar to Theorem 2 for the first

_log

~V

_digits,

but such a result

yields

no

_improvement

of the lower bounds of the mean

values considered in

_Corollary

2.

_Therefore,

we don’t state

_it,

but we are

going

to state a

_{stronger result,}

which follows

similarly

but under

_stronger

assumptions, namely

that _q is a

_prime:

Theorem 3. Let a &#x3E; 2 be an

integer

and _{p E} P a

prime

with

(a, p) -

1.

-1,

0 i s. Furthermore let _{e, q} be

arbitrary

_positive

real numbers and

A,

(N), A2 (N)

positive

integer-valued functions

with

Then we have

for

a

positive

real number

A,

as N --~ _oc,

Again

we have the

following

_simple

_consequence:

Corollary

3. Let _{a, p,} and _eo be as in Theorem 3 and E an

arbi-trary positive

real number. Then zue

_have,

as N --~ _oo,

and

_consequently

and

The _{paper is}

_organized

as follows: in Section

_2,

we _prove Theorem 2 and

in Section 3 Theorem 3. In the final

_section,

we make some remarks.

2. Proof of the Theorem 2

In this

_section,

we use the

following

notation: with a and _q we denote two

integers

both &#x3E; 2. We define a :=

_109a

_q and assume that a is irrational.

First,

we need the well-known

_concept

of

discrepancy

(see

[6]):

Definition 1. Let be a sequences of real numbers and N &#x3E; 1. Then

the N-th

_{discrepancy of}

the _sequence xn is

defined by

where is the characteristic

_{function of}

the set

_{[a, b).}

Our first Lemma is a famous

inequality

for the

discrepancy,

which is due

Lemma 1. Let be a _sequence

of

real numbers and N &#x3E; 1. Then we have

for any

positive integer

K. The constant c is absolute.

Next,

we need a

_result,

which is a

_special

case of a more

_general

result

due to Kiss and

_Tichy

_[8].

The

_proof

follows

_{by using}

the Erdôs-Turàn

inequality

together

with Baker’s theorem on linear forms in

_logarithm.

Lemma 2. There exists a

positive

real

_{constant y}

such that

The last

_ingredient

is a _very

simple fact,

but it is one of the

key

ideas of

the

_proofs

of Theorem 2 and Theorem 3.

Lemma 3. Let n E N and we consider the _q-ary

digital expansion

(1)

of

n. Let eo be a

digital

block and

_put

m =

E’=o

eiqi.

Then

for

all

k &#x3E; s we have

Now,

we are able to _prove Theorem 2.

Proof of

Theorem 2.

_{Let _ â , l}

_N’

be a

positive

_{integer, where q}

is the constant in Lemma

_3,

and

_put

m =

eiqi.

We consider

_

It _{is easy} to see that

where K

= K - c

log

l -

s + 1 with a suitable constant c and I is either or

[0,

1[,

where

1 ’ 2 ’ _{2 1} ’ ’

Applying

Lemma 3

_yields

Next,

we consider

and it is an _easy calculation that

and

Therefore,

we have

In the sum on the left hand side of

(9),

we count all

_tuples

_{(n, k),1}

n N,

s k

_K,

such that the

_following

condition holds

where 1 is an

_integer

with 1 1

[Nt].

If we fix _n,

_then,

the above

inequality

implies

and Theorem 2 follows from

_(9).

n

’

3. Proof of Theorem 3

In this section a &#x3E; 2 is an

_{integer and p}

E P denotes a

_prime

with

(a, p)

= 1.

Let k be a

positive

integer.

With

T(Pk),

we denote the

multiplicative

order of a mod

p~‘.

For

T(P)

we

write just

T.

If p is

odd

then,

we denote

by /3

the smallest number such that 1.

_{If p = 2}

_then,

we set 8 = 1

if a = 1 mod 4 and 6 = 2 if a - 3 mod 4. In this case

#

is the smallest

Lemma 4. Let _{a, p} and

₍₃

be as above. For all

_{integers n}

we have

Proof.

See

_[9].

0

For the

_proof

of Theorem

_3,

we need estimations for

special

exponen-tial sums. The first Lemma is a

special

case of a

result,

which is due to

Niederreiter

_[13].

Lemma 5. Let k &#x3E;

_{2, h}

be

_integers

and

_{(h, p)}

= 1. Assume that

r(pk) =

Then it

_follows

The next result is due to Korobov

_{(see [10]}

or

[11]).

Lemma 6. Let m &#x3E;

_{2, h}

be

_integers

with

_{(a, m)}

= 1 and

(h, m)

= 1. Let

T be the

multiplicative

order

of a

mod m. Then me have

_for

1 N T

We will

_apply

this Lemma for the

_special

case m =

p k.

Notice that this

lemma

_{provides only}

a

_good

estimation when N is not too small. We also need

_good

estimations for very small N. The best known result in this direction is

_again

due to Korobov

_{(see [10]}

or

[11]).

Lemma 7. Let k &#x3E;

_{1, h}

be

_integers

and

_{(h, p)}

= 1. Then

for

all

_integers

N with N

_r(pk)

we have

where q

&#x3E; 0 is an absolute constant and the

zmplied

constant

depends only

on a and _p.

If n is a

_positive

_integer

then we write in the

following

for the _p-ary

digital

expansion

of a’~:

We _prove now the

following

Lemma:

Lemma 8. Let eses-l ... eo be a

_digital

block to base _p. Let _f,11 &#x3E; 0 be

We consider

Then we have

for

an

_arbitrary

_positive

real number

A,

as

and this holds

_uniformly

_for

k with

_(11).

Proof.

Put m =

EL

We use

(8)

and obtain

With the definition of

_discrepancy

₍₆₎

it follows

where the

_implied

O-constant is 1. In order to

_get

the desired

_result,

we

have to estimate the

_discrepancy

on the

right

hand side.

Therefore,

we use once more

inequality

(7).

Let 1 6 2 be a real number. We

_distinguish

between two cases.

First we consider k with

Let À &#x3E; 0 be a real number and h

10gÀ

N be a

positive

integer.

First,

we

observe for

_large

_enough

N

where

_~3

is the

_integer

introduced in the

_beginning

of the section. We use

Lemma 5 and it follows

if N is

_large

_enough.

Because of

₍₁₄₎

we can estimate the

exponential

sum

in

_inequality

₍₇₎

with

_help

of Lemma 8 for h

_10gÀ

N. It follows

where c

depends

only

on a, p and y is absolute. With

(13)

we can estimate

the

_right

hand side of the above

_inequality

Now we can finish the

proof

of the first case. We consider

and choose K =

[loga

N] .

Then,

with the estimation of the

exponential

sum, we have

where the

_implied

constant does not

_depend

on k with

(13).

_By

(12)

this

completes

the

_proof

of the first case.

Next,

we consider

Let À and h be as in the first case. With the notations of Lemma 5 and

because of

₍₁₅₎

we have for

large enough

N

It follows from Lemma 6 that the

_exponential

sum in the

_inequality

(7)

is

0,

if we sum over a

period.

Hence,

we can use the estimation of Lemma 7:

Using

(15)

it is an _easy calculation to show that

where ô is a suitable constant. Notice that the

_implied

constant does not

depend

on k.

The rest of the

_proof

of the second case is similar to the first case. If we

combine the two _cases, then we

_get

the claimed result. 0

Theorem 3 is an _{easy consequence} of this Lemma:

Proof of

Theorem 3. Of course the

_following

_equality

is true

where

_Ak

is as in Lemma 9.

4. Remarks

Remark 1. In Theorem

_2,

we consider the last

digits

of the

digital

expan-sion of the

_exponential

_sequence. Notice that the

_leading

term

_logq

N is

_exactly

the

_expected

_term,

if one assumes that the

_digits

are

equidis-tributed.

A similar result should hold for more

_{digits. However,}

with the method

of

_proof,

it doesn’t seem to be

_possible

to _{extend the range of}

_digits

in order to prove a

_stronger

result.

Remark 2. In Theorem

_3,

we are interested in the first

_digits

of the

digital expansion

of the

_exponential

_sequence. The result is of the same

type

as Theorem

2,

especially

we have the

expected

order of

magnitude.

Truncation of the first

_digits

is _necessary, because the

_{multiplicative}

order of a mod

p~

can be _very

small,

for small k and

therefore,

it is

possible

that

not all

_digits

occur at the k - th

_position.

_However,

the lower bound for the

digit

range could be reduced to

_{c logp logp}

N + d,

where c and d are suitable

constants,

but then A in the error term would not be

_arbitrary

_any more.

If we assume that _{p is} not _necessary a

_prime,

then the method of

_proof

could be used to

_get

a result for the first

log

N

digits

of the

digital

_expansion.

In this situation

_only

the

_simpler

estimation of Lemma 7 for the involved

exponential

sum of the form

is needed.

These

_exponential

sums have been _very

frequently

studied,

because

they

are

_important

in the

_theory

_{of generating}

_{pseudo-random}

numbers with the

linear

_congruential

_generator

_(see

for instance

_[12]

or

[13]).

The

_proof

of Theorem 3

_{heavily depends}

on estimations of these

expo-nential sums,

especially

one needs estimations for _very short intervals. Of

course, better estimations would

yield

a better

_result,

_however,

to obtain

good

estimations for _very short intervals seems to be a hard

_problem.

Remark 3. In the

_proof

of Theorem

_1,

all

_digits

of the

_digital

_expansions

are considered. One can

_adopt

this idea to

_get

a lower bound for the

number of

_digits,

which are not zero and therefore a lower bound for the mean value of the

sum-of-digits

function.

_However,

we have not been able to obtain a lower bound better than the one in Theorem 1 with such ideas.

It seems that for better results

_{by taking}

all

_digits

into

_account,

a

_totally

new method is needed.

We end with a

conjecture,

which seems to be far _away from what can be

Conjecture

1. Let a, q &#x3E; 2 be

integers

and assume that

_log~

_q is irrational.

Let _eo be a

_digital

block. Then we have

As a _consequence one would have N as lower bound for the mean values

in

_Corollary

2 and

_Corollary

3.

Acknowledgement.

The author would like to thank Prof. Drmota and Prof. Niederreiter for valuable

_suggestions

and

_helpful

discussions about this

_topic.

References

[1] A. BAKER, Transcendental number theory. Cambridge University Press, Cambridge - New York - Port Chester - Melbourne - _Sydney, 1990.

[2] A. _BAKER, G. _WÜSTHOLZ, Logarithmic forms and group varieties. J. Reine Angew. Math.

442 _(1993), 19-62.

[3] G. _BARAT, R. F. _TICHY, R. _{TIJDEMAN, Digital} blocks in linear numeration _systems. In Number _Theory in _{Progress (Proceedings} of the Number _Theory Conference _{Zakopane 1997,} K. _Gyôry, H. _Iwaniec, and J. Urbanowicz _edt.), de Gruyter, Berlin, New York, 1999,

607-633.

[4] N. L. _BASSILY, I. _KÀTAI, Distribution _of the values _{of q-additive functions} on _polynomial

sequences. Acta Math. Hung. 68 _(1995), 353-361.

[5] R. BLECKSMITH, M. FILASETA, C. _NICOL, A result on the _{digits of} an. Acta Arith. 64 _(1993), 331-339.

[6] M. DRMOTA, R. F. TICHY, Sequences, Discrepancies and _{Applications.} Lecture Notes Math.

1651, Springer, 1997.

[7] P. _ERDÖS, P. _TURÁN, On a _problem in the _{theory of uniform} distributions _{I, II. Indagationes} Math. 10 _{(1948), 370-378,} 406-413.

[8] P. KISS, R. F. TICHY, A _{discrepancy problem} with _applications to linear recurrences _I,II.

Proc. Japan Acad. Ser. A Math. Sci. 65 _(1989), no. _{5, 135-138;} no. _6, 191-194.

[9] N. M. KOROBOV, Trigonometric sums with _{exponential functions} and the distribution _of

signs in repeating decimals. Mat. Zametki 8 _(1970), 641-652 = Math. Notes 8 (1970), 831-837.

[10] N. M. KOROBOV, On the distribution of digits in _{periodic fractions.} Matem. Sbornik 89

(1972), 654-670.

[11] N. M. KOROBOV, Exponential sums and their _{applications.} Kluwer Acad. _Publ.,

North-Holland, 1992.

[12] H. _{NIEDERREITER,} On the Distribution _of Pseudo-Random Numbers Generated _by the Linear

Congruential Method II. Math. _Comp. 28 _(1974), 1117-1132.

[13] H. _{NIEDERREITER,} On the Distribution _of Pseudo-Random Numbers Generated by the Linear

Congruential Method III. Math. _Comp. 30 _(1976), 571-597. Michael FucHs Institut für Geometrie TU Wien Wiedner Hauptstrasse 8-10/113 A-1040 Wien Austria E-mail : _{fuchsogeometrie.tuvien.ac.at}