COLLOQUIUM MATHEMATICUM

VOL. 84/85 2000 PART 2

A Z ^d GENERALIZATION OF THE DAVENPORT–ERD ˝ OS CONSTRUCTION OF NORMAL NUMBERS

BY

MORDECHAY B. L E V I N (RAMAT-GAN)

MEIR S M O R O D I N S K Y (TEL-AVIV)

Dedicated to the memory of the late Anzelm Iwanik, a friend and fellow mathematician

Abstract. We extend the Davenport and Erd˝ os construction of normal numbers to the Z

^case.

1. Introduction. A number α ∈ (0, 1) is said to be normal to the base b if in the b-ary expansion of α, α = .d 1 d 2 . . . (d i ∈ {0, 1, . . . , b − 1}, i = 1, 2, . . .), each fixed finite block of digits of length k appears with an asymptotic frequency of b ^−k along the sequence (d _i ) _i≥1 . Normal numbers were introduced by Borel [B]. Champernowne [C] gave an explicit construc- tion of such a number, namely

θ = .123456789101112 . . .

obtained by successively concatenating all the natural numbers written to base 10.

Let ϕ(x) = αx ^r +α ₁ x ^r−1 +. . .+α _r−1 x+α _r (α > 0, r ≥ 1) be a polynomial with integer coefficients such that ϕ(n) ≥ 0 (n = 1, 2, . . .). Davenport and Erd˝ os [DE] generalized Champernowne’s construction and proved that the number

.ϕ(1)ϕ(2) . . . ϕ(n) . . .

obtained by successively concatenating the b-expansions of the numbers ϕ(n) (n = 1, 2, . . .) is also normal. We refer the reader to other generalizations of Champernowne’s construction which appear in [AKS] and [SW].

In [LeSm] we extend Champernowne’s construction to Z ^d , d > 1, ar- rays of random variables, which we shall call Z ^d -processes. We shall deal with stationary Z ^d -processes, that is, processes with distribution invariant

2000 Mathematics Subject Classification: Primary 11K16, 28D15.

Work supported in part by the Israel Science Foundation Grant No. 366-172.

[431]

under the Z ^d action. We shall call a specific realization of a Z ^{d -process a}

“configuration”.

In this note we generalize the Davenport and Erd˝ os construction to the Z ^d case. For the sake of clarity, we carry out the proof only for the case d = 2. The generalization for general d > 2 is easy and straightforward. We begin with a very simple generalization (see also [Ci] and [KT]).

We denote by N the set of non-negative integers. Let d, b ≥ 2 be two integers, N ^{d = {(n} 1 , . . . , n d ) | n i ∈ N , i = 1, . . . , d}, ∆ b = {0, 1, . . . , b − 1}, Ω = ∆ ^N _b

.

We shall call ω ∈ Ω a configuration (lattice configuration). A configura- tion is thus a function ω : N ^{d → ∆ b .}

Given a subset F of N ^{d , ω F} will be the restriction of the function ω to F . Let N ∈ N ^{d , N = (N} 1 , . . . , N d ). We denote a rectangular block by

F N = {(f 1 , . . . , f d ) ∈ N ^{d | 0 ≤ f} i < N i , i = 1, . . . , d},

h = (h 1 , . . . , h d ), h i ≥ 1, i = 1, . . . , d; G = G h is a fixed block of digits G = (g i ) i∈F

, g i ∈ ∆ b , χ ω,G (f ) is the characteristic function of the block of digits G shifted by the vector f in the configuration ω:

χ _ω,G (f ) = n 1 if ω(f + i) = g _i , ∀i ∈ F _h , 0 otherwise.

(1)

Definition. ω ∈ Ω is said to be rectangular normal if for any h ∈ N ^d and block G _h ,

#{f ∈ F _N | χ _ω,G (f ) = 1} − b ^−h

^...h

N ₁ . . . N _d = o(N ₁ . . . N _d ) (2)

as max(N ₁ , . . . , N _d ) → ∞.

As remarked, in what follows we shall consider the case d = 2.

Construction. ^{The map}

L(f ₁ , f ₂ ) =  f ₁ ² + f 2 if f 2 < f 1 , f ₂ ² + 2f 2 − f 1 if f 2 ≥ f 1 , (3)

is a bijection between N ^and N ² , inducing a total order on N ^{2 from the} usual one on N ^{. Let I n = [α −1/(2r) b 2n}

2

/r ], n = 1, 2, . . . We define the con- figuration ω n on F _(2nI

_,2nI

₎ as the concatenation of I _n ² 2n × 2n blocks of digits with the lower left corner (2nx, 2ny), 0 ≤ x, y < I n . To each of these blocks we assign the number ϕ(L(x, y)). Next we use the b-expansion of ϕ(L(x, y)) according to the order L to obtain the digits of the 2n × 2n block considered. It is easy to obtain an analytic expression for the digits of the configuration ω n :

ω n (2nx + s, 2ny + t) = a _L(s,t) (u), (4)

where

u = u(x, y) = ϕ(L(x, y)),

(5)

s, t, x, y are integers, 0 ≤ x, y < I n , 0 ≤ s, t < 2n, and n = X

i≥0

a i (n)b ⁱ (6)

is the b-expansion of the integer n.

Next we define inductively a sequence of increasing configurations ω _n on F _(2nI

_,2nI

₎ . Put ω ₁ ⁰ = ω ₁ , ω _n+1 ⁰ (f ) = ω ⁰ _n (f ) for f ∈ F _(2nI

_,2nI

₎ and ω _n+1 ⁰ (f ) = ω _n+1 (f ) otherwise. Put

ω _∞ = lim ω ⁰ _n , (ω _∞ ) F

= ω ⁰ _n , n = 1, 2, . . . (7)

Theorem. ^ω ∞ is rectangular normal.

The proof of the Theorem is given in Section 3.

2. Auxiliary notation and results. Let (u _x ) _x≥0 be an arbitrary se- quence in [0, 1). The quantity

D(N ) = D((u x ) ^{N −1} _x=0 ) = sup

γ∈(0,1]

    1

N #{0 ≤ n ≤ N − 1 | u x ∈ [0, γ)} − γ     (8)

is called the discrepancy of (u x ) ^{N −1} _x=0 . The sequence (u x ) x≥0 is said to be uniformly distributed in [0, 1) if D(N ) → 0.

To estimate the discrepancy we use the Erd˝ os–Tur´ an inequality (see, for example, [DrTi], p. 15)

N D(N ) ≤ 3 2

 2N

H + 1 + X

0<|m|≤H

| P N −1

x=0 e(mu x )|

m

 , (9)

where e(y) = e ^2πiy , m = max(1, |m|) and H ≥ 1 is arbitrary.

We shall use the following Weyl inequality (see, for example, [DrTi], p. 15):

L

X

x=1

e(ψ(x))  

 ≤ C(θ)L ^1+θ (q ⁻¹ + L ⁻¹ + qL ^−k ) ²

, (10)

where ψ(x) = βx ^k + β 1 x ^k−1 + . . . + β k−1 x + β k , |β − p/q| < 1/q ² , (p, q) = 1 and θ > 0 is arbitrary.

3. Proof of the Theorem. Consider the configuration ω n , where n satisfies the following inequality:

2(n − 1)I _n−1 ≤ max(N ₁ , N ₂ ) < 2nI _n . Let h 1 , h 2 ≥ 1 be integers and

d _i

_,i

∈ {0, 1, . . . , b − 1}, 0 ≤ i ₁ < h ₁ , 0 ≤ i ₂ < h ₂ .

We consider the block of digits G = (d i

,i

) 0≤i

<h

, 0≤i

<h

, the configuration

ω n , and the block of digits ω 0 = (ω n (i, j)) 0≤i<N

, 0≤j<N

.

To compute the number of appearances of the block G in the configura- tion ω 0 , we introduce the following notation (see (1)):

V n,G (L 1 , M 1 ; L 2 , M 2 ) (11)

= [

(i,j)∈[L

,L

+M

)×[L

,L

+M

)

{(i, j) | χ ω

,G (i, j) = 1}, V _n,G (N ₁ , N ₂ )= V _n,G (0, N ₁ ; 0, N ₂ ).

(12) Let

N 1 = 2nN 11 + N 12 , N 2 = 2nN 21 + N 22 , with N 12 , N 22 ∈ [0, 2n).

(13)

Next, we fix s, t ∈ [0, 2n), and compute the number of appearances of G in the configuration ω ₀ = (ω _n (i, j)) _0≤i<N

_{, 0≤j<N}

such that the shift of the block G by the vector (i, j) satisfies i ≡ s (mod 2n), j ≡ t (mod 2n). Set

A s,t,G (M 1 , M 2 ) = [

(i,j)∈[0,2nM

)×[0,2nM

)

{(i, j) | χ ω

,G (i, j) = 1 and (14)

i ≡ s, j ≡ t (mod 2n)}.

Let ε > 0 be arbitrary. To complete the proof of the Theorem it is sufficient to prove that for all s, t ∈ [εn, 2n(1 − ε)),

|#A s,t,G (M 1 , M 2 ) − b ^−h

^h

M 1 M 2 | < εM 1 M 2 . Observe that

V _n,G (N ₁ , N ₂ ) = V _n,G (2nN ₁₁ , 2nN ₂₁ ) ∪ V _n,G (0, 2nN ₁ ; 2nN ₂₁ , N ₂₂ ) (15)

∪ V n,G (2nN 11 , N 12 ; 0, N 2 ) and

V n,G (2nN 11 , 2nN 21 ) = [

0≤s<2n

[

0≤t<2n

A s,t,G (N 11 , N 21 ), (16)

V _n,G (0, 2nN ₁₁ ; 2nN ₂₁ , N ₂₂ ) (17)

= [

0≤s<2n

[

0≤t<N

(A _s,t,G (N ₁₁ , N ₂₁ + 1) \ A _s,t,G (N ₁₁ , N ₂₁ )).

Now let

v(i ₁ , i ₂ ) = v(s, t, i ₁ , i ₂ ) = L(s + i ₁ , t + i ₂ ).

(18)

Everywhere below 0 ≤ s, t < 2n − h 1 h 2 . Using (4)–(6) we see that the condition

ω n (2nx + s + i 1 , 2ny + t + i 2 ) = d i

,i

, ∀(i 1 , i 2 ) ∈ [0, h 1 ) × [0, h 2 ), (19)

is equivalent to the statement

a _v(i

_,i

₎ (u(x, y)) = d i

,i

, ∀(i 1 , i 2 ) ∈ [0, h 1 ) × [0, h 2 ).

(20)

From (14), (1) and (19), (20) we obtain A _s,t,G (M ₁ , M ₂ )

(21)

= {(2nx + s, 2ny + t) ∈ [0, 2nM 1 ) × [0, 2nM 2 ) |

a _v(i

_,i

₎ (u(x, y)) = d i

,i

, ∀(i 1 , i 2 ) ∈ [0, h 1 ) × [0, h 2 )}.

Let k ₁ , . . . , k _h (h = h ₁ h ₂ ) be an increasing sequence of integers from the set {v(s, t, i 1 , i ₂ ) + 1 | i ₁ = 0, 1, . . . , h ₁ − 1, i 2 = 0, 1, . . . , h ₂ − 1}, (22)

and µ(i 1 , i 2 ) ∈ [1, h] ((i 1 , i 2 ) ∈ [0, h 1 ) × [0, h 2 )) be a sequence of integers so that

µ(i 1 , i 2 ) > µ(j 1 , j 2 ) ⇔ v(s, t, i 1 , i 2 ) > v(s, t, j 1 , j 2 ), (23)

where i ν , j ν ∈ [0, h ν ), ν = 1, 2. It is evident that

k _µ(i

_,i

₎ = v(s, t, i ₁ , i ₂ )+1, i ₁ = 0, 1, . . . , h ₁ −1, i ₂ = 0, 1, . . . , h ₂ −1.

(24) Now put

d µ(i

,i

) = d i

,i

, i 1 = 0, 1, . . . , h 1 − 1, i 2 = 0, 1, . . . , h 2 − 1.

(25)

From (21)–(25) we see that

A s,t,G (M 1 , M 2 ) = {(2nx + s, 2ny + t) ∈ [0, 2nM 1 ) × [0, 2nM 2 ) | (26)

a _k

₋₁ (u(x, y)) = d _i , ∀i ∈ [1, h ₁ h ₂ ]}.

Lemma 1. ^{Let M 1 , M 2 ∈ [0, I n ), I n = [α −1/(2r) b 2n}

^/r ], s, t ∈ [0, 2n − 15h], h = h 1 h 2 . Then

#A s,t,G (M 1 , M 2 ) (27)

=

b

−1

X

x

=0

. . .

b

−1

X

x

=0

B _st (M ₁ , M ₂ , d(x ₂ , . . . , x _h )), where

B _st (M ₁ , M ₂ , d) = #{(x, y) ∈ [0, M ₁ ) × [0, M ₂ ) | (28)

{u(x, y)b ^−k

} ∈ [d/b ^k

^−k

⁺¹ , (d + 1)/b ^k

^−k

⁺¹ )}, and

d = d(x 2 , . . . , x h ) (29)

= d ₁ + x ₂ b + d ₂ b ^k

^−k

+ . . . + x _h b ^k

^−k

⁺¹ + d _h b ^k

^−k

.

P r o o f. From (6), we see that the condition a k

−1 (u(x, y)) = d i , ∀i ∈ [1, h], is equivalent to the statement

u(x, y) = x 1 + d 1 b ^k

⁻¹ + x 2 b ^k

+ d 2 b ^k

⁻¹ + . . . + x h b ^k

+ d h b ^k

⁻¹ + x h+1 b ^k

,

with x i ∈ [0, b ^k

^−k

⁻¹ ), k 0 = 0, i = 1, 2, . . . , h, x h+1 ≥ 0. Using (26) and

(29) we get

A s,t,G (M 1 , M 2 ) (30)

=

b

−1

[

x

=0

. . .

b

−1

[

x

=0

{(2nx + s, 2ny + t) ∈ [0, 2nM 1 ) × [0, 2nM 2 ) | u(x, y) = x 1 + d(x 2 , . . . , x h )b ^k

⁻¹ + x h+1 b ^k

} for arbitrary integers x 1 ∈ [0, b ^k

⁻¹ ), x h+1 ≥ 0. Bearing in mind that the condition

u(x, y) = x 1 + db ^k

⁻¹ + x h+1 b ^k

is equivalent to the condition

{u(x, y)b ^−k

} ∈

 d

b ^k

^−k

⁺¹ , d + 1 b ^k

^−k

⁺¹



we deduce from (30) and (28) that A _s,t,G (M ₁ , M ₂ )

=

b

−1

[

x

=0

. . .

b

−1

[

x

=0

{(2nx + s, 2ny + t) ∈ [0, 2nM 1 ) × [0, 2nM 2 ) |

{u(x, y)b ^−k

} ∈ [d/b ^k

^−k

⁺¹ , (d + 1)/b ^k

^−k

⁺¹ )}.

Lemma 2. ^{Let 1 ≤ M 2 ≤ M 1 ∈ [b ξ2n}

^{/r , I n ), I n = [α −1/(2r) b 2n}

^{/r ], ξ =} (1 − ε) ² + ε ∈ (0, 1), s, t ∈ [εn, 2n(1 − ε)], h = h 1 h 2 , n ≥ 4/ε ² , ε ∈ (0, 1/(4r)) and 0 < |m| ≤ H = b ^k

^−k

^+s+t . Then

S(m) =

M

−1

X

y=0 M

−1

X

x=0

e(mu(x, y)b ^−k

) = O(M 1 M 2 H ⁻¹ b ⁻ⁿ

^ε

²

).

(31)

P r o o f. By (22), (18) and the condition of the lemma, we get k 1 = max(s ² + t, t ² + t − s), k 1 − s − t > ε ² n ² /2, (32)

0 ≤ k _h − k ₁ ≤ 2sh ₁ + 2th ₂ + 2h ² ₁ + 2h ² ₂ ≤ 8nh + 4h ² , (33)

H = O(b ^16nh ).

Let

M 0 ∈ [b ^ξ

²ⁿ

^/r , I n ], ξ 1 = (1 − ε) ² + ε ² , (34)

and

σ(y) =

M

−1

X

x=0

e(mϕ(x ² + y)b ^−k

).

Applying Weyl’s inequality (10) with θ = ε ² r2 ^−2r−2 , L = M 0 , k = 2r,

β = αmb ^−k

, q = b ^k

/d and d = gcd(b ^k

, αm), where α > 0 is an integer, we

obtain

|σ(y)|

(35)

≤ C(ε ² r2 ^−2r−2 )M ₀ ^1+ε

^r2

(b ^−k

d + M ₀ ⁻¹ + b ^k

d ⁻¹ M ₀ ^−2r ) ²

. Using the assumption of the lemma, (34), (18), (22) and (32), (33) we get

b ^−k

d ≤ b ^−k

α|m| ≤ αb ^−k

H = αb ^−k

^+s+t (36)

= O(b ^−k

^/2 ) = O(b ^−ε

ⁿ

^/2 ), M ₀ ⁻¹ ≤ b ^{−2((1−ε)}

^+ε

⁾ⁿ

^/r < b ⁻ⁿ

^/r , (37)

b ^k

d ⁻¹ M ₀ ^−2r ≤ b ^(k

⁾

(M 0 ) ^−2r _min ≤ b ⁴ⁿ

^(1−ε)

+2n−2r(((1−ε)

+ε

)2n

/r)

(38)

= b ⁻⁴ⁿ

^ε

⁺²ⁿ = O(b ⁻²ⁿ

^ε

).

Now from (33)–(38) we have

M ₀ ⁻¹ σ(y) = O(M ₀ ^ε

^r2

b ^−ε

ⁿ

²

) = O(b ^−ε

ⁿ

²

), and

HM ₀ ⁻¹ σ(y) = O(b ^−ε

ⁿ

²

).

(39) Putting

σ 1 =

M

−1

X

x=0

e(mϕ(x)b ^−k

), (40)

σ 2 =

M

−1

X

y=0 M

−1

X

x=0

e(mϕ(x ² + y)b ^−k

), (41)

σ 3 =

M

−1

X

x,y=0

e(mϕ(x ² + y)b ^−k

), (42)

and using (5) and (31), we obtain S(m) − σ 1 =

M

−1

X

y=0 M

−1

X

x=M

e(mu(x, y)b ^−k

) = σ 2 − σ 3 . (43)

If M 2 < b ^ξ

²ⁿ

^/r , we apply (39) with M 0 = M 1 for σ 2 , and the trivial estimate for σ 1 and σ 3 :

HM ₁ ⁻¹ M ₂ ⁻¹ S(m) = O(b ^−ε

ⁿ

²

+ (HM ₁ ⁻¹ M ₂ ⁻¹ )M ₂ ² ) (44)

= O(b ^−ε

ⁿ

²

+ b ^16nh+(ξ

^−ξ)2n

^/r )

= O(b ^−ε

ⁿ

²

).

Now let M ₂ ≥ b ^ξ

²ⁿ

^/r . We apply (39) with M ₀ = M ₂ for σ ₂ and for σ ₃ : HM ₁ ⁻¹ M ₂ ⁻¹ (|σ 2 | + |σ 3 |) = O(b ^−ε

ⁿ

²

).

(45)

To estimate the sum σ 1 we apply Weyl’s inequality with θ = ε ² r2 ^−2r−3 , L = M ₂ ² , k = r, β = αmb ^−k

, q = b ^k

/d, d = gcd(b ^k

, αm), and repeat the calculations (35)–(39):

HM ₁ ⁻¹ M ₂ ⁻¹ |σ 1 | = O(b ^−ε

ⁿ

²

).

(46)

By (44)–(46) the assertion of the lemma follows.

Lemma 3. Under the assumptions of Lemma 2,

D = D(({u(x, y)b ^−k

}) ^M _{x=0, y=0}

^{−1, M}

⁻¹ ) = O(b ^k

^−k

^−s−t ).

(47)

P r o o f. We apply Lemma 2, (31), (33) and Erd˝ os–Tur´ an’s inequality with H = b ^k

^−k

^+s+t to get

D = O



H ⁻¹ + (M ₁ M ₂ ) ⁻¹ X

0<|m|≤H

|S(m)|

m



= O

 H ⁻¹



1 + 1

s + t + 1 X

0<|m|≤H

1 m



= O(H ⁻¹ (1 + (s + t + 1) ⁻¹ log H))

= O(H ⁻¹ (1 + (s + t + 1) ⁻¹ (k h − k 1 + s + t))) = O(H ⁻¹ ).

Using the definition of discrepancy (8), we get:

Corollary 1. Under the assumptions of Lemma 2, B st (M 1 , M 2 , d) = M 1 M 2 b ^k

^−k

⁻¹ (1 + O(b ^−s−t )), (48)

where B st (M 1 , M 2 , d) is defined in (28).

Corollary 2. Under the assumptions of Lemma 2,

#A _s,t,G (M ₁ , M ₂ ) = b ^−h M ₁ M ₂ + O(M ₁ M ₂ b ^−s−t ).

(49)

P r o o f. This follows from (28), Lemma 1 and Corollary 1.

Lemma 4. Under the assumptions of Lemma 2, let 1 ≤ N 2 ≤ N 1 ∈ [2nb ^ξ2n

^/r , 2nI n ). Then

#V _n,G (N ₁ , N ₂ ) − b ^−h 4n ² N ₁ N ₂ = 200ε ₀ N ₁ N ₂ + O(N ₁ N ₂ /n), |ε 0 | ≤ ε.

P r o o f. Using (16) we have V _n,G (2nN ₁₁ , 2nN ₂₁ ) (50)

= [

εn≤s,t<2n(1−ε)

[

min(s,t)<εn

[

2n(1−ε)≤max(s,t)<2n

A _s,t,G (N ₁₁ , N ₂₁ ).

We apply (49) to the first union and the trivial estimates to the other unions:

#V n,G (2nN 11 , 2nN 21 ) (51)

= X

εn≤s,t<2n(1−ε)

(b ^−h N 11 N 21 + O(N 11 N 21 b ^−s−t )) + 16ε 1 n ² N 11 N 21

= b ^−h 4n ² N 11 N 21 + 32ε 2 n ² N 11 N 21 + O(N 11 N 21 ),

N ₂₁ ≥ 1, |ε i | < ε, i = 1, 2.

Similarly, from (17) and (49) we obtain (52) #V _n,G (0, 2nN ₁₁ ; 2nN ₂₁ , N ₂₂ )

= X

0≤s<2n

X

0≤t<N

(b ^−h N 11 + O(N 11 b ^−s−t )) + 16ε 3 nN 11 N 22

= b ^−h 2nN ₁₁ N ₂₂ + 32ε ₄ nN ₁₁ N ₂₂ + O(N ₁₁ N ₂₂ ) 5 with |ε i | < ε, i = 3, 4. We get a trivial estimate from (11)–(13):

#V _n,G (2nN ₁₁ , N ₁₂ ; 0, N ₂ ) ≤ N ₂ N ₁₂ ≤ 2nN ₂ < N ₁ N ₂ /n.

Now the assertion of the lemma follows from (13), (15), and (51)–(52).

Similar notation is introduced for the configuration ω (instead of ω n ):

V G (P 1 , P 2 ) = {(v 1 , v 2 ) ∈ [0, P 1 ) × [0, P 2 ) | (53)

ω(v ₁ + i ₁ , v ₂ + i ₂ ) = d _i

_,i

, ∀(i ₁ , i ₂ ) ∈ [0, h ₁ ) × [0, h ₂ )}.

We prove the Theorem for the case N 1 ≥ N 2 . The other case is simi- lar.

End of the proof of the Theorem. Let 1 ≤ N 2 ≤ N 1 , N 1 ≥ 4b ⁸ . Then there exists n ≥ 3 so that

N 1 ∈ [2(n − 1)I n−1 − h, 2nI n − h).

(54) Now let

N ₁ ⁰ = 2(n − 1)I n−1 − h, N ₂ ⁰ = min(N 2 , N ₁ ⁰ ).

(55)

From (53) and the definition of the configurations ω, ω n we get

#V _G (N ₁ , N ₂ ) = #V _n,G (N ₁ , N ₂ ) − #V _n,G (N ₁ ⁰ , N ₂ ⁰ ) + #V _G (N ₁ ⁰ , N ₂ ⁰ ) (56)

+ 2ε 1 hN ₂ ⁰ + 2ε 2 N 1 min(h, N 2 − N ₂ ⁰ )

with |ε _i | ≤ 1, i = 1, 2. It is easy to see that if N ₂ ≤ n, then N ₂ = N ₂ ⁰ , otherwise h ≤ hN ₂ /n and

#V G (N 1 , N 2 ) − #V n,G (N 1 , N 2 ) = #V G (N ₁ ⁰ , N ₂ ⁰ ) − #V n,G (N ₁ ⁰ , N ₂ ⁰ ) (57)

+ 4ε 3 hN 1 N 2 /n with |ε 3 | ≤ 1.

Analogously

#V G (N ₁ ⁰ , N ₂ ⁰ ) − #V n,G (N ₁ ⁰ , N ₂ ⁰ ) = #V G (N ₁ ⁰⁰ , N ₂ ⁰⁰ ) − #V n−1,G (N ₁ ⁰⁰ , N ₂ ⁰⁰ ) (58)

+ 4ε 4 hN 1 N 2 /n,

where

N ₁ ⁰⁰ = 2(n − 2)I n−2 − h, N ₂ ⁰⁰ = min(N 2 , N ₁ ⁰⁰ ), |ε 4 | ≤ 1.

(59)

It is evident that

#V G (N ₁ ⁰⁰ , N ₂ ⁰⁰ ) + #V n,G (N ₁ ⁰⁰ , N ₂ ⁰⁰ ) ≤ 2N ₁ ⁰⁰ N ₂ ⁰⁰ < 2N 1 N 2 /n.

(60)

From (56)–(60) we obtain

#V G (N 1 , N 2 ) = #V n,G (N 1 , N 2 ) − #V n,G (N ₁ ⁰ , N ₂ ⁰ ) + #V n−1,G (N ₁ ⁰ , N ₂ ⁰ ) + O(N 1 N 2 /n).

It is easy to verify that

b ^ξ2n

^/r = o(I n−1 ),

where ξ = (1 − ε) ² + ε ∈ (0, 1), and I _n = [α ^−1/(2r) b ²ⁿ

^/r ]. Hence N ₁ ∈ [2nb ^ξ2n

^/r , 2nI _n ) and we can apply Lemma 4:

#V _G (N ₁ , N ₂ )

= b ^−h N 1 N 2 − b ^−h N ₁ ⁰ N ₂ ⁰ + 400ε 5 N 1 N 2 + b ^−h N ₁ ⁰ N ₂ ⁰ + O(N 1 N 2 /n)

= b ^−h N ₁ N ₂ + 400ε ₅ N ₁ N ₂ + O(N ₁ N ₂ /n) with |ε ₅ | ≤ ε.

Now from (1), (2), and (53) we obtain the assertion of the Theorem.

Acknowledgments. We are grateful to the referee for his corrections and suggestions.

REFERENCES

[AKS] R. A d l e r, M. K e a n e and M. S m o r o d i n s k y, A construction of a normal number for the continued fraction transformation , J. Number Theory 13 (1981), 95–105.

[B] E. B o r e l, Les probabilit´ es d´ enombrables et leurs applications arithm´ etiques , Rend. Circ. Mat. Palermo 27 (1909), 247–271.

[C] D. J. C h a m p e r n o w n e, The construction of decimals normal in the scale ten , J. London Math. Soc. 8 (1933), 254–260.

[Ci] J. C i g l e r, Asymptotische Verteilung reeller Zahlen mod 1 , Monatsh. Math.

64 (1960), 201–225.

[DE] H. D a v e n p o r t and P. E r d ˝ o s, Note on normal numbers , Canad. J. Math. 4 (1953), 58–63.

[DrTi] M. D r m o t a and R. T i c h y, Sequences , Discrepancies and Applications , Lec- ture Notes in Math. 1651, Springer, 1997.

[KT] P. K i r s c h e n h o f e r and R. F. T i c h y, On uniform distribution of double sequences , Manuscripta Math. 35 (1981), 195–207.

[LeSm] M. B. L e v i n and M. S m o r o d i n s k y, On explicit construction of normal

lattices , preprint.

[SW] M. S m o r o d i n s k y and B. W e i s s, Normal sequences for Markov shifts and intrinsically ergodic subshifts , Israel J. Math. 59 (1987), 225–233.

Department of Mathematics and Computer Science Bar-Ilan University

52900 Ramat-Gan, Israel E-mail: mlevin@macs.biu.ac.il

School of Mathematical Sciences Tel Aviv University 69978 Tel-Aviv, Israel E-mail: meir@math.tau.ac.il

Received 27 August 1999; (3825)

revised 17 February 2000