### J

### OURNAL DE

### T

### HÉORIE DES

### N

### OMBRES DE

### B

### ORDEAUX

### M

### ORDECHAY

### B. L

### EVIN

### On normal lattice configurations and simultaneously

### normal numbers

### Journal de Théorie des Nombres de Bordeaux, tome

### 13, n

o_{2 (2001),}

### p. 483-527

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

© Université Bordeaux 1, 2001, tous droits réservés.

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

### On normal

### lattice

_{configurations }

### and

### simultaneously

### normal

### numbers

par MORDECHAY B. LEVIN

Dedicated to Professor Michel Mendès France

in the occasion _{of }his 65th _{birthday}

RÉSUMÉ. Soient q, q1, ... , qs > 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

_{(à }

un facteur _{logarithmique }

_{près) }

avec la borne inférieure
de la

_{discrépance }

des suites ordinaires dans un cube de
dimension s

_{(s,M,N }

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

Nous calculons aussi une borneinférieure de la

_{discrépance }

_{(à }

un facteur _{logarithmique }

### près)

dela suite

_{(problème }

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

_{(up }

to a _{logarithmic }

### factor)

with the lower bound of the### discrepancy

of_{ordinary }

sequences in s-dimensional unit
cube

_{(s,M,N }

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

We also find a lower bound of the### discrepancy

### (up

to a_{logarithmic factor) }

of the sequence ### ({03B11qn1},

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

### (Korobov’s

### 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}

### expansion

of_{cx, }cx -

### dl d2 ...

### (d2

_{E }

### {0,1, " ’

q -

### 1},

i =### 1, 2, - ~ - ),

each fixed finite block of_{digits }

of _{length }

1~ _{appears }with an

_{asymptotic }

_{frequency}

of

_{along }

the _{sequence }Normal numbers were introduced

### by

Manuscrit reçu le 2 mars 1999.

Borel

_{(1909). }

_{Champernowne }

_{(1935) }

_{gave }an

_{explicit }

construction of such
a

_{number, }

_{namely}

obtained

_{by successively concatenating }

all the natural numbers.
1.1.1. We denote

_{by }

N the set of _{non-negative }

_{integers. }

Let _{d, q > }

2 be two
Y II -...

We shall call w e Q a

_{confaguration }

### (lattice

### configuration ).

A_{configuration}

is a function w : :

### Nd

- A. Let### h, N

### E N‘ d ,

_{7 }h =

### (h 1, ... , hd ) ,

N =### (Nl, ... , Nd).

We denote a_{rectangular }

block _{by}

G =

### Gh

is a fixed block of_{digits }

G = E _{[0, hj), j =}

### 1,...,d~.

1.1.2.

_{Definition. }

A lattice _{configuration, }

w E _{Q, }

is said to be normal
### (rectangular normal)

if for_{any }h E

### I~d

with_{h1 ~ ~ ~ hd > }

1 and block of
### digits Gh,

where i =

### (ii, ... ,

### id),

and### max(Nl, ... ,

### Nd) ~

_{oo.}

It is evident that almost _{every } is normal. The constructive

_{proof }

of
the existence of the normal lattice _{configuration }

is _{given }

in _{~LS1~, (LS2~.}

### Below,

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 =

### ~0,71)

x ~ ~ ~ x_{[0,1’8) }

be a box in ### [0, 1);

and### Av(N)

be anumber of indexes n E

_{[1, N] }

such that xn lies in v. The _{sequence }

### (xn)

issaid to be

_{uniformly }

distributed in _{[0, 1) }

if for every box v,
7i " ’ 7s- The

### quantity

is called the

_{discrepancy }

of
It is known

_{(Roth, }

_{[Ro]) }

that for _{any sequence in }

_{[0,1)~,}

1.1.4. The double _{sequence } E

_{[0,1)8, }

_{(n, m }

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

is said tobe

_{uniformly }

distributed _{(Cigler, }

_{[Ci]) }

if
Kirschenhofer and

_{Tichy }

_{[KiTi] }

_{investigated }

double sequences over finite
sets

_{(see }

also references in _{[DrTi, p.364], [KN, }

_{p. }

_{18]).}

1.2. It is known

_{(Wall, 1949) }

that a number a is normal to the base _{q}

if and

_{only }

if the _{sequence }is

_{uniformly }

distributed in _{[0, 1) }

_{(see}

### [KN,

p.### 70]).

It is easy to prove### similarly

### (see

### Appendix

of this### paper)

the### following

statement:### Proposition

1. Let_{q > }

2 be _{integer, }

E _{{0, 1, ... , }

_{q - }

_{1}, }

_{m, n }=

### 1, 2, ....

The lattice_{configuration }

normal _{if }

and _{only if }

_{for}

all s > 1 the double _{sequence}

is

_{uniformly }

distributed in _{~0,1)S, }

where
1.2.1. In

_{[Le3] }

it was _{proved }

_{explicitly }

that there exists a normal number
a with

The estimate of

_{discrepancy }

was _{previously }

known ### O(N -2/3 log4/3

### N) (see

### [Ko2],[Le2]).

### According

to_{(3), }

the estimate _{(6) }

cannot be _{improved }

### essen-tially.

Our

_{goal }

is to find a lower bound of ### discrepancy

of the doublese-quence

### (4).

The main idea of the paper is the### using

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

### (see (5)

and### (9)).

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

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

with_{optimal }

coefficient
### (see

### [Ko3]).

We_{provide }

the _{following }

construction of a normal lattice
### con-figuration :

1.2.2 Construction. Let _{Pl, p2 }be distinct

_{primes; }

_{(q, plp2) }

=
where

Theorem 1. There exist

_{integers }

## aT ~~

_{(m, r - }

_{1, 2, ... , v }

= ### 0,1 )

### satisf

### y-ing

### ( 11 ),

such that_{f or }

all _{s, N, M > }

1 we have
with

_{max(M, N) -+ }

_{oo, }and the constant

_{implied by }

0 ### only depends

on s.We note that

_{according }

to _{(3), }

the estimate _{(12) }

cannot be _{improved by}

more than the _{power }of the

_{logarithmic multiplier.}

### Corollary.

Let s,### q >

2. There exist numbers_{al, }... , as

### (simultaneously

normal to the base

_{q) }

such that
The

_{discrepancy }

estimate was ### previously

known as### O(N-’I’)

### [Kol]

and### O(N-2/310gs+2

### N)

### [Le2].

1.3. Let _{s, ql, ... , }

_{qs > }

2 be _{integers. }

Numbers _{al, ... , as }are said to be

### simultaneously

normal to the base_{(ql, ... , qs) }

_{[Kol],[Ko3] }

if the _{sequence}

is

_{uniformly }

distributed in _{[0,1)~.}

In

_{[Kol], }

Korobov obtained the first _{examples }

of _{simultaneously }

normal
numbers _{using }

normal _{periodic }

_{systems, }

_{completely }

_{uniformly }

distributed
sequences, and estimates of ### trigonometric

sums with_{exponential }

functions
### (see

also_{[Ko3]). }

In _{[Kol], }

Korobov constructed _{simultaneously }

normal
numbers with
and

_{posed }

the _{problem }

of _{finding }

_{simultaneously }

normal numbers with
a maximum

### decay

of the### discrepancy

of the_{sequence }

### (13).

In### [Lel],

### si-multaneously

normal numbers with### DN

=### N)

werecon-structed. Here we find

### simultaneously

normal numbers with thediscrep-ancy estimate We note that

### according

to### (3),

thises-timate cannot be

_{improved by }

more than the _{power }of the

### logarithmic

### multiplier.

1.3.1 Construction. Let _{p }be

_{prime; }

_{(qi, p) }

_{= 1, i = 1, ... , s;}

where

_{{}

Theorem 2. Let s > 2. There exist

_{integers}

We _{prove }this theorem in Section 4. Theorem 1 is

_{proved }

in Section 3.
Section 2 contains _{auxiliary }

results.
2.

_{Auxiliary }

results
### First,

some further notation is_{necessary. }For

### integers

d > 1### 2,

let

_{Cd(l) }

be the set of all nonzero lattice ### points

### (hI,..., hd) E Zd

withand

for h =

### (hi,

... ,

### hd)

E### Cd(l).

For real### t,

the abbreviation### e (t) =

isused.

_{Subsequently, }

four known results are ### stated,

which follow from### [Ko3,

Lemma

_{2], [Ei, }

Lemma _{3], [Ko3, }

_{p.13, Ni, }

p. ### 35]

and### [Ni,

Theorem### 3.10],

Lemma 1.

_{Let p > }

_{2, a }

be _{integers,}

Then

Lemma 2. Let

_{q > }

2 be an ### integer.

Then### for

any divisor v### of

_{q }with 1 v

_{q.}

Lemma 3. Let

_{A, B, T }

be _{integers,}

### According

to_{[}

the

_{proof }

of Lemma 3 _{repeats }

that of _{[Ko3, p.13~. }

0
### Applying

this lemma_{twice, }

we _{get}

Lemma 4. Let N > 1 and P > 2 be

_{integers. }

Let ### tn =

_{yn/P }

E ### [0, 1)d

with yn E_{{O, 1,... , }

P - _{for }

0 n N. Then the _{discrepancy of }

the
### points to,

### tl, ... , tN_1

### satisfies

where

where

and

Hence

### Similarly

to_{[Ni, }

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

### Bearing

in mind that_{max(}

we find that

Lemma 6. Let

It _{is easy }to see that

and

### Applying

### (2),

we obtain:and and

From

_{(20) }

and _{(21) }

we _{get :}

where ’ I

where

_{cp(x) }

is a Euler ### function,

It is evident that

Now lE

Lemma 7. Let

### Proof.

We follow_{[Ko3, }

p. ### 191].

We denote the left side of### (25)

### by

al.where

This

_{yields }

that there E _{[0, 2s - 1] }

such that
### If >

= v, thenNow let

_{A 0 }

v.
We find for

_{integers }

_{vl, v2 }and v3

### (with (vi, plp2)

=### 1,

i =### 1, 2),

thatwhere

and

Hence

From

_{(27) }

we _{see, }that

### Now,

### applying (29)

for the case of_{p,=v }and

### (30),

### (31)

for the case of### p, =f:. v,

we_{get}

We obtain from

_{(26) }

that
### Applying

Lemma_{2, }

we _{get}

### Now,

from_{(33) }

we obtain :
Let

Lemma 8. With the notation

_{defined }

above we have:
### Proof.

It follows from Lemma 7 and_{(35) }

that
where v E

_{[0, }

_{tm) ,}

Hence
and

### Changing

the order of the_{summation, }

we find that
### Now,

from_{(36), }

we obtain the assertion of the lemma.
Put

and for v E

_{~1, s - 1)}

Lemma 9. Let I

and

### Proof.

We will_{prove }the statement

_{(41). }

The _{proof }

of _{(40) }

_{repeats }

that
of _{(41). }

We denote the left side of _{(41) }

_{by }

_{Changing }

the order of the
### summation,

from_{(39) }

we _{get}

where

Let , and let

Then there is i

It _{is easy }to see that

where

We find that if 1

Now let

_{(mod}

and

where

It is _{easy }to

_{verify }

that
and

### According

to_{(7), }

we have
### Bearing

in mind that_{(}

we obtain from ### (46)

that### Now,

### (45)

and_{(43) }

_{imply }

that
### Applying

Lemma_{2, }

we _{get}

Lemma 10. Let

and

Then

### Changing

the order of the summation we obtain that### Now,

from_{(50) }

we obtain the desired result. _{Using }

_{(49), }

we ### similarly

obtain### (51)

for the case of v = 1. 0### Now,

from Lemmas 8 and 10 we_{get:}

### Corollary

4. LetThen

Lemma 11. With the notation

_{defined }

_{above, }

we have:
### Proof.

Letand let = 1 for some

_{io E }

### [0~-1].

We denote the left side of### (55)

### bya. Changing

the order of the summation we_{get }

from ### (54)

where

It _{is easy }to see that

where

It is evident that if

_{f fl }

0 _{(mod }

_{p2 ), }

then _{W (l~_2, h) }

= 0.
Let

_{f - }

0 _{(mod }

_{f }

=
### Bearing

in mind that =### 1,

we obtain

### Equation

### (57)

shows thatThen

_{equation }

_{(56) }

_{implies }

that
Now from

_{(38) }

we obtain the assertion of the lemma.
### Corollary

5. Let### Proof.

### Changing

the order of the_{summation, }

from Lemma 11 we have
We denote the

_{right }

side of _{(60) }

_{by }

Q. ### Bearing

in mind### (8)

and thatwe find that

### Taking

into account_{(49), (52), }

and that _{b~"2~ ¢ }

_{Qm,3, }

we deduce that
### Now,

from_{(58) }

and _{(60), }

we obtain the desired result.
Lemma 12. Let s . Then

We denote the left side of

_{(62) }

_{by Q. }

_{Changing }

the order of the _{summation,}

we obtain from

### (61)

thatwhere

We now obtain

It _{is easy }to see that if

### f fl

0 mod_{(pf), }

then 11
### Bearing

in mind that_{(}

we have

Hence

### Now,

from_{(63) }

and _{(39), }

we obtain the assertion of the lemma.
Then

...

### 1}.

Hence### Bearing

in mind_{(50), (52), }

and that _{b(m) 0 }

_{~3, }

we deduce that the left
side of _{(67) }

is less than _{(m }

+ _{2)/(8(m }

+ _{1)) ~1/4. }

Then from _{(64) }

we
obtain the desired result. D

Now we choose vectors

_{(m }

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

for the construction in### (9)

inthe

_{following }

_{way:}

### According

to_{(53),}

We take

_{arbitrarily }

from the set
### Taking

into account_{(53),(59), }

and _{(65), }

we obtain that the set
is not

_{empty. }

Let _{a(l),..., }

be chosen. Then we choose
### arbi-trarily

so thatThe _{sequence }of vectors

_{(m }

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

is constructed_{inductively.}

Next we fix s > _{1, }

and we consider _{integers }

m such that s.
Main Lemma. With the notation

_{defined above, }

we have:
where

### Proof.

The_{proof }

follows from _{(7), (8), }

_{(36), }

_{(52), (58), (64), (68) }

and
### (69).

D3. Proof of Theorem 1.

In this

_{section, }

the _{integer }

s is fixed. For m = ### 1, 2, ... ,

letand let

It is easy to see that is

_{rectangular }

domain _{(i }

= ### 1, 2)

and t### Thus,

to_{compute }

_{DGm,,, }

it is sufficient to find the estimate _{of DF, }

where F
is an _{arbitrary }

_{rectangular }

domain in _{vm, }

m =
### l, 2, ....

### According

to_{(9),}

the

_{analytic expression }

of _{depends }

on the _{position }

of _{(x, y) }

in _{Ym.}

We will consider three _{possibilities }

for the _{position }

of F in _{Vm: }

the _{middle,}

the _{right }

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

### f (x, y)

E F_{I }

i mod _{(21~~), }

_{y * j }

mod _{(tm) I }

of _{rectangular}

domain _{F, }

and we will _{compute }

the _{discrepancy }

on the sub-domain _{Fi,j }

for
First we obtain a

### simple

_{expression }

for _{fayqxl, }

where _{(x, y) }

_{belongs }

to a
middle domain of

_{V",,:}

### Proof.

From_{(8) }

we obtain
### Let (;

Then

_{equation }

_{(8) }

_{implies }

that _{r(y) }

= _{m.}

### According

to_{(9), }

we find that
### where y

and### Bearing

in mind thatwe obtain that

Now

, - - -

- _

### equation

### (77)

and the conditions of the lemma show that### r(y)

m. From### (9)

we_{get}

where 0 is an

### integer.

It is easy to see### that,

### here,

Hence we can

_{repeat }

the calculations ### (77) - (79).

Thus we obtain the desiredresult. 0

Define

_{G, }

=
### Proof.

Let_{(x, y) }

E _{Gi ,}

and let

### (We

continue_{periodically }

the coordinates of vector ### a~~~.)

From Lemma 13we have that

We denote the left side of

_{(82) }

_{by u. }

_{Applying }

Lemma 5 with k ### = I~~i~

=### andqi=q

### Using

Lemma 6 with, we obtain from

### (84)

Then

_{Corollary }

3 _{(with j }

.J
where : and

It is easy to see that

where

_{Ml pm - }

and _{M2p2 T }

_{1 (modpm) . }

_{Taking }

into account
that

_{XIP2 }

+ _{ylp’ }

passes the ### complete

residue### system

and### using

Lemma_{1, }

we have that
Now from

_{(24),(35) }

and _{(86) }

we find that
### Applying

### (7), (81)

and the condition of the_{lemma, }

we obtain the desired
result. D

We can now use Lemma 14 to

_{compute }

the _{discrepancy }

of the considered
double sequence in the ### rectangular

domain E C### V.

where the constant

_{implied by }

0 _{only depends }

on s.
where

_{K, K2, K3, K5, L, L2, L5 }

> 0 are _{integers, }

and
It is evident that

### Using

### (2)

and_{(75), }

we obtain
From

_{(81) }

we obtain :
We obtain from

_{(89),(88),(7), }

and _{(8) }

that
### Applying

### (75), (70),

and Lemma_{14, }

we _{get }

from ### (7)

and### (91)

that### Similarly,

estimates are valid for the cases of sets### E2

and### E4.

Thus- . -

Consider the

_{right }

bourne of _{Vm . }

Define
### Proof.

Let_{(x, y) }

E _{G2. }

Thus _{y }

_{tmp2:, }

and from _{(77) }

we find that
### r(y)

m.### Similarly

to_{(78) }

and _{(80) }

we have that
Hence

with and

### apply

### (94)

for the_{pair }

_{(x, y }

+ _{i). }

_{Bearing }

in mind _{(83), }

we obtain
### with

and xo =

### (PI -

### 1)p-.

We denote the left side of### (93)

### by

~.### Applying

obtain from

_{(19)}

### Using

Lemma 6 with iwe obtain from

### (95)

Here we now

### replace

fractions with denominators### PIP2

and### pr+1pr+l

with fractions with denominator

_{p2 +1, }

and we _{again apply }

Lemma 5 with
the _{parameters }2s instead of _{s, p2 }instead of _{qi, }and m + 1 instead of

### k~z~

Then

_{Corollary }

2 _{(}

that
### implies

### Taking

into account_{(7), (54), (92), }

Lemma _{1, }

and the conditions of the
lemma we obtain :
Thus we obtain the assertion of the lemma.

It is evident that

### Using

### (2)

and_{(75), }

we find that
It follows from

_{(75) }

that
where

_{G2 - }

_{G2 (m, }

_{X3, }

_{y4, }

_{L2, }

_{L3), }

and z = ### K6 -

### K4 km .

### Bearing

inmind that

_{(98) }

is _{true, }

we can _{apply }

Lemma 16:
Now from

_{(71), (99), }

and _{(100) }

we obtain the assertion of the ### corollary.

We now consider the _{upper }bourne of

### form=

### 1,2,...

Then

where : 1

### Proof.

Let_{(x, y) }

E _{G3 }

and let i E _{(0, t,,t - }

_{y2). }

The _{equality }

_{(83) }

and the
conditions of the lemma _{show, }

that _{Bm(Y2, i) }

= 0 . The ### pair

### (x,

_{y }+

### i)

satisfies the conditions of Lemma 13. The

_{equation }

_{(84) }

_{implies }

that
Now let i E

_{[tm - }

y2, s - ### 1].

Then y + i > and_{r(y }

+ _{i) }

= m + 1
### (see (8)

and_{(77) ). }

The _{pair }

_{(~,?/+z) }

satisfies the conditions of Lemma 13
### (with

m + 1 instead of_{m). }

_{Hence, }

we have from ### (84)

and### (104)

thatwhere

### Using

Lemma 6 withobtain from

_{(103), }

and _{(105)}

where Xl = + +

Here we now

_{replace }

fractions with denominators _{PIP2 }

and
the

_{parameters }

2s instead of _{s, PI }instead of

_{qi, }and m + 1 instead of i - I where for and for Then

_{Corollary}

### implies

thatWe obtain from

_{(61) }

that
Thus we derive the desired result.

### Proof.

LetIt _{is easy }to see that

If

_{K3 f 0, }

then
Now let

_{I~3 }

> 0. Put
It is evident that

### Using

### (2)

and_{(75), }

we find that
From

_{(7) }

and _{(8) }

we obtain :
It follows from

_{(75) }

and _{(101), }

that
where

_{G3 }

= ### G3(m,

_{XS, X6, Y2, }

### K2,

### K3) .

Because

_{(107), }

we can ### apply

Lemma 17. Thus from### (72), (108),

and### (109),

we obtain

### (106).

0### Now,

### combining

### (87),(97),

and_{(106) }

we obtain :
Lemma 18. Let

_{Kl, K6, }

_{Li, L6 > }

0 be _{integers }

and E =
### K6)

x### [tmll,

### L6)

C Then### Proof.

Letand let

It is _{easy }to see that

Now

_{equations }

_{(7) }

and _{(75) }

_{imply }

that
From Lemma

_{15, }

_{Corollary }

_{7, }

and _{Corollary 8, }

we _{get}

From

_{(110), }

we obtain the assertion of the lemma. 0
End of the

_{proof }

of Theorem 1. We use notations ### (73), (74),

and### (75).

Let

### It

is evident thatand z is the

_{rectangular }

domain ### (

be coordinates of the vertex of

### Gm) :

_{2}

From

_{(7), (8), }

and _{(48), }

we see that
Hence

### Using

### (2), (75),

and_{(111), }

we find that
Thus we obtain the assertion of Theorem 1. 0

Remark.

_{Using }

_{(76), (95), }

and _{(103) }

we can find ### specifically digits

### (i, j

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

of_{a }normal lattice

### configuration

### (see (5)).

With the_{}

no-tations defined

_{above, }

if _{(x, y) = }

+ + _{x3, tr,.tyl }

+ _{y2) E }

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

### Proof.

We will_{prove }the

_{inequality }

_{(114). }

The _{proof }

of _{(115) }

_{repeats }

that
of _{(114). }

We denote the left side of _{(114) }

_{by }

Q. ### Changing

the order of the### summation,

from_{(112) }

we obtain
where

### Bearing

in mind that_{(}

and
we find that

### Corollary

9. There exist_{integers}

and

We use such

_{integers }

l
numbers _{aI, ... , as }

_{(15).}

to construct the real

### Proof.

We denote the left side of_{(118) }

_{by }

0152. ### Equation

### (2)

### implies

that 7_ 1 1with

We

_{apply}

to Lemma 5:
where _{q }=

Then Lemma 6

_{(with }

_{ki = 1~?.,z - }

X3 ### and li

= =_{1, ... , }

### s)

shows thatNow

_{Corollary }

2 _{(with }

T = P = ### pm)

### implies

that### Applying

Lemma_{1, }

_{(14), }

and _{(112), }

we obtain
From

_{(14), }

_{(116), (120), }

and _{Corollary }

_{9, }

we _{get }

### (118).

_{Using }

### (117),

we### similarly

obtain_{(119). }

0
End of the

_{proof }

of Theorem 2. Let N be in Define
### D(N1,

### N2)

= 0 for### 0,

and### Using

### (2), (14),

and Lemma 20 we have for M E that### Applying

### (2),(14),

### (118),

### (119), (121),

and_{(122), }

we _{get }

the assertion of
Theorem 2:

5.

_{Appendix}

The

_{proof }

of _{Proposition }

1. We follow _{[KN p.70]. }

Let _{~1~2 > }

1 be
a

### configuration

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

### uniformly

distributed in_{[0,1)81. }

Then
and so cv is a normal lattice

### configuration.

### Conversely,

if cv is a normal### configuration,

thenwhere >

holds for all

_{G 81 ,S2. }

Therefore
for all boxes with

Now let v be a box with

### arbitrary Ii E

### (0, 1] (i

=_{1, ... , }

### sl ),

and letf E

_{(o,1) }

be _{given. }

Put s2 = and ### put hi

= =### 1,...

_{,81.}

Then

where

It is _{easy }to see that

### According

to_{[Ni, }

Lemma 3.9 _{]}

### max(i mes

vl-rnes### Imes

v2-rraes### vI) :

From

_{(123) }

and _{(124) }

we deduce that
where

_{jcij I }

and _{max(M, N) -- }

oo . Hence for all 1 and all boxes
v C _{[0,1)~, Av (M, N) - }

MNmes v + _{o(MN), with max(M, N) - }

oo,
and so is a

_{uniformly }

distributed double
, _

sequence in

### 0,1 ) s 1.

D### Acknowledgment.

I am_{very }

_{grateful }

to the referee for _{many }corrections and

_{suggestions }

which _{improved }

this _{paper.}

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Mordechay B. LEVIN

Department of Mathematics and Computer Science Bar-Ilan University

Ramat-Gan, 52900 Israel