# On normal lattice configurations and simultaneously normal numbers

## Full text

(1)

o

### p. 483-527

<http://www.numdam.org/item?id=JTNB_2001__13_2_483_0>

© Université Bordeaux 1, 2001, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

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

(2)

### numbers

par MORDECHAY B. LEVIN

Dedicated to Professor Michel Mendès France

in the occasion of his 65th birthday

RÉSUMÉ. Soient q, q1, ... , qs &#x3E; 2 des entiers et 03B11, 03B12, ... des nombres réels. Dans cet

### article,

on montre que la borne inférieure

de la

### discrépance

de la suite double

coïncide

un facteur

### près)

avec la borne inférieure

de la

### discrépance

des suites ordinaires dans un cube de

dimension s

=

### 1, 2,...) .

Nous calculons aussi une borne

inférieure de la

un facteur

de

la suite

de

### Korobov) .

ABSTRACT. Let q, q1, ... , qs ~ 2 be

### integers,

and let 03B1i , 03B12 , ... be a sequence of real numbers. In this paper we prove that the

lower bound of the

### discrepancy

of the double sequence

coincides

to a

### factor)

with the lower bound of the

of

### ordinary

sequences in s-dimensional unit

cube

=

### 1, 2, ... ) .

We also find a lower bound of the

to a

of the sequence

## ..., {03B1sqns})Nn=1

### problem) .

1. Introduction.

1.1. A number a E

### (0, 1)

is said to be normal to the base q, if in a q-ary

of cx, cx -

E

q -

i =

### 1, 2, - ~ - ),

each fixed finite block of

of

### length

1~ appears with an

of

### along

the sequence Normal numbers were introduced

### by

Manuscrit reçu le 2 mars 1999.

(3)

Borel

gave an

### explicit

construction of such

a

obtained

### by successively concatenating

all the natural numbers.

1.1.1. We denote

N the set of

Let

### d, q &#x3E;

2 be two

Y II -...

We shall call w e Q a

A

### configuration

is a function w : :

- A. Let

7 h =

N =

We denote a

block

G =

### Gh

is a fixed block of

G = E

1.1.2.

A lattice

w E

### Q,

is said to be normal

if for any h E

with

1 and block of

where i =

and

### Nd) ~

oo.

It is evident that almost every is normal. The constructive

### proof

of the existence of the normal lattice

is

in

to

### simplify

the calculations we consider

### only

the case of d = 2.

1.1.3. Let

### (x~)

be an infinite sequence of

### points

in an s-dimensional unit

cube =

x ~ ~ ~ x

be a box in

and

### Av(N)

be a

number of indexes n E

### [1, N]

such that xn lies in v. The sequence

is

said to be

distributed in

### [0, 1)

if for every box v,

7i " ’ 7s- The

is called the

of

It is known

### [Ro])

that for any sequence in

### [0,1)~,

(4)

1.1.4. The double sequence E

=

is said to

be

distributed

### [Ci])

if

Kirschenhofer and

### investigated

double sequences over finite

sets

### (see

also references in

p.

1.2. It is known

### (Wall, 1949)

that a number a is normal to the base q

if and

### only

if the sequence is

distributed in

p.

### 70]).

It is easy to prove

of this

the

statement:

1. Let

2 be

E

q -

m, n =

The lattice

normal

and

### for

all s &#x3E; 1 the double sequence

is

distributed in

where

1.2.1. In

it was

### explicitly

that there exists a normal number

a with

The estimate of

was

known

to

the estimate

cannot be

Our

### goal

is to find a lower bound of

of the double

se-quence

### (4).

The main idea of the paper is the

### using

of small discrep-ancy sequences on the multidimensional unit cube to construct the

se-quence of reals

and

### (9)).

Here we use a variant of Ko-robov’s s-dimensional sequences

=

with

coefficient

We

the

### following

construction of a normal lattice

### con-figuration :

1.2.2 Construction. Let Pl, p2 be distinct

### (q, plp2)

=

(5)

where

Theorem 1. There exist

## aT ~~

=

such that

all

1 we have

with

### max(M, N) -+

oo, and the constant

0

on s.

We note that

to

the estimate

cannot be

### improved by

more than the power of the

Let s,

### q &#x3E;

2. There exist numbers al, ... , as

### (simultaneously

normal to the base

such that

The

estimate was

known as

and

### [Le2].

1.3. Let s, ql, ... ,

2 be

### integers.

Numbers al, ... , as are said to be

### simultaneously

normal to the base

if the sequence

is

distributed in

In

### [Kol],

Korobov obtained the first

of

normal numbers

normal

### uniformly

distributed sequences, and estimates of

sums with

functions

also

In

### [Kol],

Korobov constructed

### simultaneously

normal numbers with

(6)

and

the

of

### simultaneously

normal numbers with

a maximum

of the

of the sequence

In

### si-multaneously

normal numbers with

=

### N)

were

con-structed. Here we find

### simultaneously

normal numbers with the

discrep-ancy estimate We note that

to

### (3),

this

es-timate cannot be

### improved by

more than the power of the

### multiplier.

1.3.1 Construction. Let p be

where

### {

Theorem 2. Let s &#x3E; 2. There exist

### integers

We prove this theorem in Section 4. Theorem 1 is

### proved

in Section 3. Section 2 contains

results.

2.

results

### First,

some further notation is necessary. For

d &#x3E; 1

let

### Cd(l)

be the set of all nonzero lattice

with

and

for h =

... ,

E

For real

the abbreviation

is

used.

### Subsequently,

four known results are

Lemma

Lemma

p.

and

Theorem

(7)

Lemma 1.

be

Then

Lemma 2. Let

2 be an

Then

any divisor v

q with 1 v q.

Lemma 3. Let

be

to

the

of Lemma 3

that of

0

this lemma

we

### get

Lemma 4. Let N &#x3E; 1 and P &#x3E; 2 be

Let

E

with yn E

P -

0 n N. Then the

the

(8)

where

where

(9)

and

Hence

to

Lemma

### 3.9],

the second sum is not more than

" "

and we obtain the first

### part

of the lemma.

. It is easy to see that

and

1

in mind that

### max(

we find that

(10)

Lemma 6. Let

It is easy to see that

and

we obtain:

and and

(11)

From

and

we

where ’ I

where

is a Euler

### function,

It is evident that

Now lE

Lemma 7. Let

We follow

p.

### 191].

We denote the left side of

al.

(12)

where

This

that there E

such that

= v, then

Now let

v.

We find for

vl, v2 and v3

=

i =

that

where

and

(13)

Hence

From

we see, that

### applying (29)

for the case of p,=v and

for the case of

we

We obtain from

that

Lemma

we

(14)

from

### (33)

we obtain :

Let

Lemma 8. With the notation

above we have:

### Proof.

It follows from Lemma 7 and

that

where v E

Hence

(15)

and

the order of the

we find that

from

### (36),

we obtain the assertion of the lemma.

Put

and for v E

Lemma 9. Let I

(16)

and

### Proof.

We will prove the statement

The

of

that of

### (41).

We denote the left side of

the order of the

from

we

### get

where

Let , and let

Then there is i

It is easy to see that

where

We find that if 1

Now let

(17)

and

where

It is easy to

that

and

to

we have

in mind that

we obtain from

that

and

that

(18)

Lemma

we

Lemma 10. Let

and

Then

(19)

### Changing

the order of the summation we obtain that

from

### (50)

we obtain the desired result.

we

obtain

### (51)

for the case of v = 1. 0

### Now,

from Lemmas 8 and 10 we

### Corollary

4. Let

Then

(20)

Lemma 11. With the notation

we have:

### Proof.

Let

and let = 1 for some

### [0~-1].

We denote the left side of

### bya. Changing

the order of the summation we

from

### (54)

where

It is easy to see that

where

(21)

It is evident that if

0

then

= 0.

Let

0

=

in mind that =

we obtain

shows that

Then

that

Now from

### (38)

we obtain the assertion of the lemma.

5. Let

(22)

the order of the

### summation,

from Lemma 11 we have

We denote the

side of

Q.

in mind

and that

we find that

into account

and that

we deduce that

from

and

### (60),

we obtain the desired result.

(23)

Lemma 12. Let s . Then

We denote the left side of

the order of the

we obtain from

### (61)

that

where

We now obtain

(24)

It is easy to see that if

0 mod

then 11

in mind that

we have

Hence

from

and

### (39),

we obtain the assertion of the lemma.

Then

...

(25)

Hence

in mind

and that

### ~3,

we deduce that the left side of

is less than

+

+

Then from

### (64)

we

obtain the desired result. D

Now we choose vectors

=

### 1, 2, ... )

for the construction in

in

the

way:

to

We take

from the set

into account

and

### (65),

we obtain that the set

is not

Let

### a(l),...,

be chosen. Then we choose

### arbi-trarily

so that

The sequence of vectors

=

is constructed

### inductively.

Next we fix s &#x3E;

and we consider

### integers

m such that s.

(26)

Main Lemma. With the notation

we have:

where

The

follows from

and

### (69).

D

3. Proof of Theorem 1.

In this

the

### integer

s is fixed. For m =

### 1, 2, ... ,

let

and let

It is easy to see that is

domain

=

and t

to

### DGm,,,

it is sufficient to find the estimate

where F is an

domain in

m =

to

the

of

on the

of

in

### Ym.

We will consider three

for the

of F in

the

the

### right

bourne and the upper bourne. Next we will consider sub-domains

E F

i mod

mod

of

domain

and we will

the

### discrepancy

on the sub-domain

### Fi,j

for

(27)

First we obtain a

for

where

to a

middle domain of

From

we obtain

Then

that

= m.

to

we find that

and

in mind that

we obtain that

(28)

Now

, - - -

- _

### (77)

and the conditions of the lemma show that

m. From

we

where 0 is an

### integer.

It is easy to see

Hence we can

the calculations

### (77) - (79).

Thus we obtain the desired

result. 0

Define

=

(29)

Let

E

and let

continue

### periodically

the coordinates of vector

### a~~~.)

From Lemma 13

we have that

We denote the left side of

Lemma 5 with k

=

Lemma 6 with

, we obtain from

Then

3

### (with j

.J

(30)

where : and

It is easy to see that

where

and

into account

that

+

passes the

residue

and

Lemma

we have that

Now from

and

we find that

### (7), (81)

and the condition of the

### lemma,

we obtain the desired

result. D

We can now use Lemma 14 to

the

### discrepancy

of the considered double sequence in the

domain E C

### V.

where the constant

0

on s.

(31)

where

&#x3E; 0 are

### integers,

and

It is evident that

and

we obtain

From

we obtain :

We obtain from

and

that

and Lemma

we

from

and

that

### Similarly,

estimates are valid for the cases of sets

and

Thus

- . -

(32)

Consider the

bourne of

Define

Let

E

Thus y

and from

we find that

m.

to

and

we have that

Hence

with and

(33)

for the

+

in mind

we obtain

and xo =

### 1)p-.

We denote the left side of

~.

obtain from

Lemma 6 with i

we obtain from

Here we now

### replace

fractions with denominators

and

### pr+1pr+l

with fractions with denominator

and we

Lemma 5 with

(34)

Then

2

that

into account

Lemma

### 1,

and the conditions of the lemma we obtain :

Thus we obtain the assertion of the lemma.

It is evident that

and

we find that

(35)

It follows from

that

where

X3, y4,

and z =

in

mind that

is

we can

Lemma 16:

Now from

and

### (100)

we obtain the assertion of the

### corollary.

We now consider the upper bourne of

Then

where : 1

Let

E

and let i E

The

### (83)

and the conditions of the lemma

that

= 0 . The

y +

### i)

satisfies the conditions of Lemma 13. The

that

(36)

Now let i E

y2, s -

### 1].

Then y + i &#x3E; and

+

= m + 1

and

The

### (~,?/+z)

satisfies the conditions of Lemma 13

we have from

and

that

where

Lemma 6 with

obtain from

and

where Xl = + +

Here we now

### replace

fractions with denominators

and

(37)

the

### parameters

2s instead of s, PI instead of qi, and m + 1 instead of i - I where for and for Then

that

(38)

We obtain from

### (61)

that

Thus we derive the desired result.

### Proof.

Let

It is easy to see that

If

then

Now let

### I~3

&#x3E; 0. Put

It is evident that

and

we find that

From

and

we obtain :

It follows from

and

that

where

=

XS, X6, Y2,

Because

we can

### apply

Lemma 17. Thus from

and

we obtain

0

and

we obtain :

Lemma 18. Let

0 be

and E =

x

C Then

(39)

### Proof.

Let

and let

It is easy to see that

Now

and

that

From Lemma

and

we

From

### (110),

we obtain the assertion of the lemma. 0

End of the

### proof

of Theorem 1. We use notations

and

Let

is evident that

and z is the

domain

### (

be coordinates of the vertex of

2

From

and

we see that

Hence

(40)

and

### (111),

we find that

Thus we obtain the assertion of Theorem 1. 0

Remark.

and

we can find

=

### 1, 2, ... )

of a normal lattice

### (see (5)).

With the

no-tations defined

if

+ +

+

### y2) E

then f-% B 4. Proof of Theorem 2 Let Lemma 19. Let Then and

(41)

### Proof.

We will prove the

The

of

that of

### (114).

We denote the left side of

Q.

the order of the

from

we obtain

where

in mind that

and

we find that

(42)

9. There exist

and

We use such

### integers

l

numbers aI, ... , as

### (15).

to construct the real

### Proof.

We denote the left side of

0152.

that 7_ 1 1

with

We

to Lemma 5:

where q =

(43)

Then Lemma 6

X3

= =

shows that

Now

2

T = P =

that

Lemma

and

we obtain

From

and

we

we

obtain

0

End of the

### proof

of Theorem 2. Let N be in Define

= 0 for

and

### (2), (14),

and Lemma 20 we have for M E that

and

we

the assertion of

Theorem 2:

5.

The

of

1. We follow

Let

1 be

(44)

a

w =

and real numbers

Now let

The block

### (

is identical with if and

### only

if

or

or

It follows that

Now suppose that the double sequence

distributed in

### [0,1)81.

Then

and so cv is a normal lattice

### Conversely,

if cv is a normal

then

where &#x3E;

holds for all

### G 81 ,S2.

Therefore

for all boxes with

(45)

Now let v be a box with

=

and let

f E

be

Put s2 = and

= =

### 1,...

,81.

Then

where

It is easy to see that

to

Lemma 3.9

vl-rnes

v2-rraes

From

and

we deduce that

where

and

### max(M, N) --

oo . Hence for all 1 and all boxes v C

MNmes v +

oo,

and so is a

### uniformly

distributed double

, _

sequence in

D

I am very

### grateful

to the referee for many corrections and

which

### improved

this paper.

References

[C] D. J. CHAMPERNOWNE, The construction of decimal normal in the scale ten. J. London Math. Soc. 8 (1935), 254-260.

[Ci] J. CIGLER, Asymptotische Verteilung reeller Zahlen mod 1. Monatsh. Math. 64 (1960), 201-225.

[DrTi] M. DRMOTA, R. F. TICHY, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer-Verlag, Berlin, 1997.

[Ei] J. EICHENAUER-HERRMANN, A unified approach to the analysis of compound pseudoran-dom numbers. Finite Fields Appl. 1 (1995), 102-114.

[KT] P. KIRSCHENHOFER, R. F. TICHY, On uniform distribution of double sequences,

Manuscripta Math. 35 (1981), 195-207.

[Ko1] N. M. KOROBOV, Numbers with bounded quotient and their applications to questions of

Diophantine approximation. Izv. Akad. Nauk SSSR Ser. Mat. 19 (1955), 361-380.

[Ko2] N. M. KOROBOV, Distribution of fractional parts of exponential function. Vestnic Moskov. Univ. Ser. 1 Mat. Meh. 21 (1966), no. 4, 42-46.

[Ko3] N. M. KOROBOV, Exponential sums and their applications, Kluwer Academic Publishers,

Dordrecht, 1992.

[KN] L. KUIPERS, H. NIEDRREITER, Uniform distribution of sequences. John Wiley, New York,

1974.

[Le1] M. B. LEVIN, On the uniform distribution of the sequence {03B103BBx}. Math. USSR Sbornik

27 (1975), 183-197.

[Le2] M. B. LEVIN, The distribution of fractional parts of the exponential function. Soviet. Math. (Iz. Vuz.) 21 (1977), no. 11, 41-47.

[Le3] M. B. LEVIN, On the discrepancy estimate of normal numbers. Acta Arith. 88 (1999),

99-111.

[LS1] M. B. LEVIN, M. SMORODINSKY, Explicit construction of normal lattice configuration, preprint.

[LS2] M. B. LEVIN, M. SMORODINSKY, A Zd generalization of Davenport and Erdös theorem

(46)

[Ni] H. NIEDERREITER, Random Number Generation and Quasi-Monte Carlo Methods. SIAM,

[Ro] K. ROTH, On irregularities of distributions. Mathematika 1 (1954), 73-79.

Mordechay B. LEVIN

Department of Mathematics and Computer Science Bar-Ilan University

Ramat-Gan, 52900 Israel

Updating...