On the discrepancy estimate of normal numbers by

LXXXVIII.2 (1999)

On the discrepancy estimate of normal numbers

by

M. B. Levin (Tel-Aviv)

Dedicated to Professor N. M. Korobov on the occasion of his 80th birthday 1. Introduction

1.1. A number α ∈ (0, 1) is said to be normal to base q if in the q-ary expansion of α, α = .d

d

. . . (d

∈ ∆ = {0, 1, . . . , q − 1}, i = 1, 2, . . .), each fixed finite block of digits of length k appears with an asymptotic frequency of q

along the sequence (d

)

. Normal numbers were introduced by Borel (1909).

1.1.1. Let (x

)

be an arbitrary sequence of real numbers. The quan- tity

(1) D(N ) = D(N, (x

)

) = sup

γ∈(0,1]

|#{0 ≤ n < N | {x

} < γ}/N − γ|

is called the discrepancy of (x

)

, where {x} = x − [x] is the fractional part of x. The sequence {x

}

is said to be uniformly distributed (u.d.) in [0, 1) if D(N ) → 0.

1.1.2. It is known that a number α is normal to base q if and only if the sequence {αq

}

is u.d. (Wall, 1949). Borel proved that almost every number (in the sense of Lebesgue measure) is normal to base q. In [G], Gal and Gal proved that

D(N, {αq

}

) = O((N

log log N )

) for a.e. α.

1.2. In [K1] Korobov posed the problem of finding a function ψ with maximum decay, such that

∃α : D(N, {αq

}

) ≤ ψ(N ), N = 1, 2, . . .

1991 Mathematics Subject Classification: 11K16, 11K38.

Work supported in part by the Israel Science Foundation Grant No. 0366-172.

[99]

He showed that ψ(N ) = O(N

) (see [K1]). The lower bound of the discrepancy for the Champernowne and Davenport–Erd˝os normal numbers was found by Schiffer [S]:

D(N, {αq

}

) ≥ K/ log N with K > 0, N = 2, 3, . . . For a bibliography on Korobov’s problem see [Po, L1].

1.3. In [L2] we proposed using small discrepancy sequences (van der Corput type sequences and {nα}

) to construct normal numbers, and announced that

ψ(N ) = O(N

log

N ).

This result is proved below. The estimate of ψ(N ) was previously known to be O(N

log

N ) (Korobov [K2] for q prime, and Levin [L1] for arbitrary integer q). We note that the estimate obtained cannot be improved essentially, since according to W. Schmidt, 1972 (see [N, p. 24]), for any sequence of reals,

N →∞

lim N D(N )/ log N > 0.

1.4. Let x = [a

(x); a

(x), a

(x), . . .] be the continued fraction expansion of x, with partial quotients a

(x). For an integer b and Q > 1 let P

a

(b/Q) denote the sum of all partial quotients of b/Q. Following [P] we prove (see Lemma 3) that there exists an integer sequence b

and a constant K > 0 with

(2)

X

X a

({b

/q

}) ≤ Km

, m = 1, 2, . . .

Theorem 1. Let

(3) α = X

m≥1

1 q

X

0≤k<q^m

 b

k q

 1 q

where b

satisfy (2),

(4) n

= 0 and n

= X

1≤r<k

rq

, k = 2, 3, . . . Then the number α is normal to base q, and

D(N, {αq

}

) = O(N

log

N ).

1.5. Let (p

)

be Pascal’s triangle:

p

= p

= 1, i = 1, 2, . . . , p

= p

+ p

, i, j = 2, 3, . . . , and (p

)

be Pascal’s triangle mod 2:

(5) p

≡ p

mod 2, i, j = 1, 2, . . .

Every integer n ≥ 0 has a unique digit expansion in base q,

(6) n = X

j≥1

e

(n)q

with e

(n) ∈ ∆ = {0, . . . , q − 1}, j = 1, 2, . . . , and e

(n) = 0 for all sufficiently large j.

Theorem 2. Let

(7) α = X

m≥1

1 q

X

0≤n<q^2m

1 q

2^m

X

i=1

d

(n) q

where

(8) d

(n) ≡ X

j≥1

p

e

(n) mod q, d

(n) ∈ ∆, i = 1, . . . , 2

, n ∈ [0, q

),

(9) n

= 0 and n

= X

1≤r<m

2

q

, m = 2, 3, . . . Then the number α is normal to base q and

D(N, {αq

}

) = O(N

log

N ).

Remark 1. We use here the sequence of 2

× 2

matrices of Pascal’s triangle mod 2. A similar result is valid for the sequence of m × m matrices of Pascal’s triangle (or m × m matrices of Pascal’s triangle mod p) but with D(N, {αq

}

) = O(N

log

N ), where α is denoted by a concatenation of blocks ω

:

α = .ω

. . . ω

. . . , where

ω

= (d

(1) . . . d

(1) . . . d

(q

) . . . d

(q

)), m = 1, 2, . . . , and

d

(n) ≡ X

j≥1

p

e

(n) mod q.

Remark 2. Let (σ

)

be any sequence of substitutions of the set ∆ = {0, 1, . . . , q − 1}. The proof of Theorem 2 does not change if in (8) we use the functions σ

(e

(n)) instead of the functions e

(n) (see [B], [N, p. 25]).

2. Proof of the theorems. Let m ≥ 1, b, i be integers, 0 ≤ i < m, (b, q) = 1,

α

= α

(b) = X

0≤k<q^m

 bk q

 1 q

, (10)

α

= [q

{α

q

}]/q

.

(11)

It is easy to see that {{bn/q

}q

} = {bn/q

}, and {α

q

} =

 bn q

 q

+

 b(n + 1) q

 1 q

+

 b(n + 2) q

 1

q

+ . . .



=

 bn q

 +

 b(n + 1) q

 1 q

+

 b(n + 2) q

 1

q

+ . . . Therefore

(12) α

=

 bn q

 +

 b(n + 1) q

 1 q

. Let N ∈ [1, mq

] be an integer, γ ∈ (0, 1],

(13) A(γ, N, (x

)) =

 #{0 ≤ n < N | {x

} < γ} for γ > 0,

0 for γ ≤ 0,

and

(14) A(γ, Q, P, (x

)) = #{Q ≤ n < Q + P | {x

} < γ}.

Hence and from (10) we obtain

A(γ, N, {α

q

}

) = A(γ, m[N/m], {α

q

}

) (15)

+ A(γ, m[N/m], N − m[N/m], {α

q

}

)

=

m−1

X

i=0

A(γ, [N/m], {α

q

}

) + θm with θ ∈ [0, 1].

Let c = [q

γ], N

∈ [1, q

] and 0 ≤ i < m. From (11) and (13) we deduce

A

 c − 1

q

, N

, (α

)



≤ A(γ, N

, {α

q

}

) (16)

≤ A

 c + 1

q

, N

, (α

)

 .

Lemma 1. Let N ∈ [1, mq

] be an integer , γ ∈ (0, 1], (b, q) = 1. Then (17) A(γ, N, {α

q

}

)

= γN + ε

 4m + 3

X

1≤N ≤q

max

N D(N, {bn/q

}

)

 , (18) A(γ, mq

, {α

q

}

) = γmq

+ 3ε

m

with |ε

| < 1, j = 1, 2.

P r o o f. Let 0 ≤ i < m, d, d

, and d

be integers, d = d

q

+ d

, d

∈ [0, q

), d

∈ [0, q

). By (12) and (13) we get

A

 d

q

, N

, (α

)



= #



0 ≤ n < N

 bn q

 +

 b(n + 1) q

 1

q

< d

q

+ 1 q

· d

q

 . Consequently,

(19) A(d/q

, N

, (α

)

) = T

(N

) + T

(N

), where

T

(N ) = #



0 ≤ n < N    

 bn q



< d

q

 , (20)

T

(N ) = #



0 ≤ n < N    

 bn q



= d

q

and

 b(n + 1) q



< d

q

 . (21)

Let N

= N

q

+ N

with N

∈ [0, q

) and N

∈ [0, q

). It is easy to see that

T

(N

) = T

(q

N

) + T

(N

).

We see from (20) and (1) that

(22) T

(N

q

) = N

d

, and

T

(N

) = d

q

N

+ εN

D

 N

,

 bn q



n≥0



with |ε| ≤ 1.

This yields

(23) T

(N

) = d

q

N

+ ε max

1≤N <q^m−i

N D

 N,

 bn q



n≥0



with |ε| ≤ 1.

Now we compute T

(N ). Let d

be an integer, d

≡ d

b

mod q

with d

∈ [0, q

), and

Y = {0 ≤ n < N

| {bn/q

} = d

/q

}.

Clearly if {bn/q

} = d

/q

, then bn ≡ d

mod q

, n ≡ d

mod q

, and

(24) Y = {d

+ rq

| 0 ≤ r < N

} with N

=

 N

− d

− 1 q

 + 1.

Combining (21) and (1) we obtain

T

(N

) = #

 n ∈ Y

 b(n + 1) q



< d

q

 (25)

= #



0 ≤ r < N

 b(d

+ 1) q

+ br

q



< d

q



= N

d

q

+ ε

N

D

 N

,

 bn q

+ θ



n≥0



with θ = b(d

+ 1)/q

, |ε

| ≤ 1.

It follows from (1) that for every real θ,

(26) D(N, {x

+ θ}

) ≤ 2D(N, {x

}

).

By (24) and (25), this yields T

(N

) =

 N

+ q

− d

− 1 q

 d

q

+ 2ε

max

1≤N ≤qⁱ

N D(N, {bn/q

}

) (27)

= N

d

q

+ ε

+ 2ε

max

1≤N ≤qⁱ

N D(N, {bn/q

}

) with |ε

| ≤ 1, j = 2, 3.

If N

= q

, then N

= q

, and N

D(N

, {bn/q

}

) = 1. Hence and from (25) and (26) we obtain

(28) T

(q

) = d

+ 2ε

with |ε

| ≤ 1.

Substituting (23) and (27) into (19), we obtain A(d/q

, N

, (α

)

)

= N

d/q

+ ε

(1 + max

1≤N <q^m−i

N D(N, {bn/q

}

) + 2 max

1≤N ≤qⁱ

N D(N, {bn/q

}

)) with |ε

| ≤ 1.

Using (16) and (15) we get A(γ, N

, {α

q

}

)

= γN

+ ε

(2 + max

1≤N <q^m−i

N D(N, {bn/q

}

) + 2 max

1≤N ≤qⁱ

N D(N, {bn/q

}

)) with |ε

| ≤ 1, and

A(γ, N, {α

q

}

) = θm + X

γ[N/m]

+ ε



2m + 3 X

i=1

1≤N ≤q

max

N D(N, {bn/q

}

)



= γN + ε



4m + 3 X

1≤N ≤q

max

N D(N, {bn/q

}

)



,

where |ε

| ≤ 1, j = 7, 8. Assertion (17) is proved. Assertion (18) follows analogously from (22) and (28).

Lemma 2. Let j ≥ 1, 1 ≤ N ≤ q

, (b, q) = 1, and a

(x) be partial quotients of {x}. Then

N D(N, {bn/q

}

) ≤ X

a

(b/q

).

For the proof of this well-known theorem, see for example [N, p. 26].

Lemma 3. There exists a constant K > 0 and integers c

∈ [0, q

) such that

X

X a

({c

/q

}) ≤ Km

, m = 1, 2, . . .

P r o o f. According to [P, p. 2144] there exist constants K

such that X

1≤c≤q^r, (c,q)=1

X a

(c/q

) ≤ K

q

r

, r = 1, 2, . . .

Therefore

(29) X

1≤c≤q^m, (c,q)=1

X

X a

({c/q

})

= X

q

X

1≤c≤q^r, (c,q)=1

X a

(c/q

) ≤ X

q

K

r

≤ K

q

m

. Let φ(q

) = #{1 ≤ c ≤ q

| (c, q) = 1} and K = K

q/φ(q). It is known that φ(q

) = q

φ(q). Now the assertion of Lemma 3 follows from (29).

Corollary. Let 1 ≤ N ≤ mq

. Then

(30) A(γ, N, {α

(b

)q

}

) = γN + O(m

).

The statement follows from (1), (2), (10), and Lemmas 1–3.

Applying (3) and (10) we get

{αq

} = {α

(b

)q

} + θq

with 0 < θ < 1 and 0 ≤ n < mq

. Hence and from (13) we have, for N ∈ [1, mq

],

A(γ − 1/q

, N − m, {α

(b

)q

}

) ≤ A(γ, N, {αq

}

)

≤ A(γ, N, {α

(b

)q

}

).

By using (30) and (14), we obtain

(31) A(γ, n

, N, {αq

}

) = γN + O(m

) with 1 ≤ N ≤ mq

. Similarly, from (18) we deduce that

(32) A(γ, n

, mq

, {αq

}

) = γmq

+ O(m).

End of the proof of Theorem 1. For every N ≥ 1 there exists an integer k such that N ∈ [n

, n

). By (4) this yields

(33) N = n

+ R with 0 ≤ R < kq

, N > (k − 1)q

, k ≤ 2 log

N.

Applying (4), (14) and (31)–(33) we obtain A(γ, N, {αq

}

) =

k−1

X

r=1

A(γ, n

, rq

, {αq

}

) + A(γ, n

, R, {αq

}

)

=

k−1

X

r=1

(γrq

+ O(r)) + γR + O(k

)

= γN + O(k

) = γN + O(log

N ).

Thus, by (1), the theorem is proved.

Proof of Theorem 2. In [So] Sobol’ proposed the use of Pascal’s triangle mod 2 to construct small discrepancy sequences (see also [F], [N]). Here we use Pascal’s triangle mod 2 to construct normal numbers.

Let P

be a sequence of a 2

× 2

matrices such that P

=

 1 1 1 0



, . . . , P

=

 P

P

P

0

 , . . .

It is easy to prove by induction that P

is the 2

×2

upper left-hand corner of Pascal’s triangle (5), and P

is a triangular-type matrix. The following lemma is proved in [BH] for Pascal’s triangle, and it is clearly valid also for Pascal’s triangle mod 2.

Lemma 4. The determinant of any n × n array taken with its first row along a row of ones, or with its first column along a column of ones in Pascal’s triangle, written in rectangular form, is one.

From (7) we have

(34) {αq

} = .d

(n)d

(n) . . . d

(n)d

(n + 1) . . . Let 1 ≤ k, i ≤ 2

and

(35) α

(n) = [{αq

}q

]/q

. It is easy to see that

(36) α

(n)

=

 .d

(n) . . . d

(n) if k + i ≤ 2

, .d

(n) . . . d

(n)d

(n + 1) . . . d

(n + 1) otherwise.

Lemma 5. Let m, k, i, B, f be integers, 1 ≤ i, k ≤ 2

, B ∈ [0, q

), f ∈ [0, q

). Then

A(f /q

, Bq

, q

, (α

(n))

) = f + 2ε with |ε| < 1.

P r o o f. Case 1. Let k + i ≤ 2

, c

∈ ∆ = {0, 1, . . . , q − 1} (j = 1, . . . , i).

We examine the system of equations

(37) d

(n) = c

, j = 1, . . . , i, n ∈ [Bq

, (B + 1)q

).

According to (8) this system is equivalent to the system of i congruences X

1≤ν≤2^m

p

e

(n + Bq

) ≡ c

mod q, j = 1, . . . , i, n ∈ [0, q

).

Applying (6) we see that e

(n + Bq

) = e

(n) + e

(Bq

), ν = 1, 2, . . . , and

(38) X

1≤ν≤i

p

e

(n)

≡ c

− X

i<ν≤2^m

p

e

(Bq

) mod q, j = 1, . . . , i, with n ∈ [0, q

). It follows from Lemma 4 that

(39) |det(p

)

| = 1.

For any c

, . . . , c

the system (38) has a unique solution (e

(n), . . . , e

(n)), and consequently there exists a unique n

∈ [Bq

, (B + 1)q

) satisfying (37).

From (36) and (37) we see that the set {α

(n) | n ∈ [Bq

, (B + 1)q

)}

coincides with {j/q

| j ∈ [0, q

)}. Hence and from (14) we have (40) A(f /q

, Bq

, q

, (α

(n))

) = f.

Case 2. Let k + i > 2

, l

= 2

− k. As in (37) and (38), the system of equations

d

(n) = c

, j = 1, . . . , l

, (41)

d

(n + 1) = c

, j = 1, . . . , i − l

, (42)

with n ∈ [Bq

, (B + 1)q

), is equivalent to the systems of congruences

(43) X

1≤ν≤i

p

e

(n)

≡ c

− X

i<ν≤2^m

p

e

(Bq

) mod q, j = 1, . . . , l

,

(44) X

1≤ν≤i

p

e

(n + 1)

≡ c

− X

i<ν≤2^m+1

p

e

((B + [(n + 1)/q

])q

) mod q, where j = 1, . . . , i − l

and n ∈ [0, q

).

Let n = n

+ n

q

with n

∈ [0, q

) and n

∈ [0, q

). It is evident that e

(n) = e

(n

) for ν = 1, . . . , l

.

The matrix P

is triangular. Hence

p

= 0 with ν > 2

− k − j = l

− j.

The system (43) is equivalent to the following system of congruences:

(45) X

1≤ν≤l₁

p

e

(n

)

≡ c

− X

i<ν≤2^m

p

e

(Bq

) mod q, j = 1, . . . , l

, where n

∈ [0, q

) and n

∈ [0, q

).

Applying (39) with i = l

shows that this system has a unique solution with (e

(n

), . . . , e

(n

)). Consequently, there exists a unique solution n

= n

∈ [0, q

) satisfying (45).

By (41) and (43) we obtain

(46) {(d

(n

+n

q

+Bq

), . . . , d

(n

+n

q

+Bq

)) | 0 ≤ n

< q

}

= {(c

, . . . , c

) | c

∈ ∆, j = 1, . . . , l

}.

Now we examine the system (44) with n

= n

the solution of (45).

Case 2.1. Let n

≤ q

− 2. Bearing in mind that

e

(n + 1) = e

(n

+ 1 + q

n

) = e

(n

+ 1) + e

(q

n

), we deduce from (44) that

X

l1<ν≤i

p

e

(q

n

)

≡ c

− X

1≤ν≤l₁

p

e

(n

+ 1) − X

i<ν≤2^m

p

e

(Bq

) mod q with j = 1, . . . , i − l

and 0 ≤ n

< q

.

Applying Lemma 4 we obtain a unique solution for this system with (e

(q

n

), . . . , e

(q

n

)).

By (42) and (44) we get

(47) {(d

(n

+ n

q

+ Bq

+ 1), . . . , d

(n

+ n

q

+ Bq

+ 1)) | 0 ≤ n

< q

} = {(c

, . . . , c

) | c

∈ ∆, j = 1, . . . , i − l

}.

Let

(48) F = {d

(n) . . . d

(n)d

(n + 1) . . . d

(n + 1) |

0 ≤ n

< q

− 1, 0 ≤ n

< q

, n = n

+ n

q

+ Bq

}, and

(49) g

= d

(q

− 1 + Bq

), ν = 1, . . . , l

. From (46) and (47) we have

(50) F = {(c

, . . . , c

) | c

∈ ∆, j = 1, . . . , i, (c

, . . . , c

) 6= (g

, . . . , g

)}

and

(51) #F = q

− q

.

Case 2.2. Let n

= q

− 1, n

∈ [0, q

− 2] and n = n

+ n

q

. Then

e

(n

+ 1) = 0 for 1 ≤ ν ≤ l

and e

(n + 1) = e

((n

+ 1)q

) for l

< ν ≤ i.

The system (44) is equivalent to the following system of congruences:

(52) X

l1<ν≤i

p

e

((n

+ 1)q

)

≡ c

− X

i<ν≤2^m

p

e

(Bq

) mod q, j = 1, . . . , i − l

, with 0 ≤ n

≤ q

− 2.

For n

∈ [0, q

− 2] we have the q

− 1 distinct vectors of (e

((n

+ 1)q

), . . . , e

((n

+ 1)q

)).

Using Lemma 4 and by (52) we obtain for n

∈ [0, q

− 2] the q

− 1 distinct vectors of (c

, . . . , c

).

Let

G = {(g

, . . . , g

, d

((n

+ 1)q

+ Bq

), . . . , d

((n

+ 1)q

+ Bq

)) | 0 ≤ n

≤ q

− 2}.

From (42), (44) and (52) we find that #G = q

− 1, and from (46) and (48)–(51) that #(F ∪ G) = q

− 1. Hence and from (36) the set {α

(n) | n ∈ [Bq

, (B + 1)q

− 2]} coincides with q

− 1 distinct values of j/q

with j ∈ [0, q

). By (14) we get

A(f /q

, Bq

, q

, (α

(n))

) = f + 2ε with |ε| < 1.

Hence and from (40) we have the assertion of Lemma 5.

Corollary 1.

(53) A(γ, Bq

, q

, {αq

}

) = γq

+ 4ε with |ε| < 1.

P r o o f. Analogously to (16), from (14) and (35) we have A

 f − 1

q

, Bq

, q

, (α

(n))



≤ A(γ, Bq

, q

, {αq

}

)

≤ A((f + 1)/q

, Bq

, q

, (α

(n))

) with f = [γq

]. By using Lemma 5 we obtain (53).

Corollary 2. Let 1 ≤ N < 2

q

. Then

A(γ, n

, N, {αq

}

) = γN + 5qε2

with |ε| < 1, (54)

A(γ, n

, 2

q

, {αq

}

) = γ2

q

+ 5ε2

with |ε| < 1.

(55)

P r o o f. Let N

= [N/2

], N

= N − 2

N

, N

= P

i=0

b

q

with b

∈ ∆,

(56) N

= 0, N

= X

i=0

b

q

, j = 1, 2, . . . , B

= N

/q

.

It is evident that B

(i = 1, 2, . . .) are integers, and N

∈ [0, 2

). As in (15) we see from (14) that

A(γ, n

, N, {αq

}

) = εN

+

2^m

X

k=1

A(γ, N

, {αq

}

), and

A(γ, N

, {αq

}

)

=

2^m

X

i=1

A(γ, N

, b

q

, {αq

}

)

=

2

X

A(γ, N

, b

q

, {αq

}

)

=

2

X

b

X

A(γ, N

+ Bq

, q

, {αq

}

).

Using (56) we have A(γ, n

, N, {αq

}

)

= ε2

+

2^m

X

k=1 2

X

i=0 b

X

A(γ, (B

+ B)q

, q

, {αq

}

).

Applying (53) we obtain

A(γ, n

, N, {αq

}

) = ε2

+

2^m

X

k=1 2

X

i=0 b

X

(γq

+ 4ε

) = γN + 5qε

2

with |ε

| ≤ 1.

Assertion (54) is proved. We prove (55) analogously.

End of the proof of Theorem 2. For every N ≥ q there exists an integer k such that N ∈ [n

, n

). By (9), this yields N = n

+R with 0 ≤ R < 2

q

, N ≥ 2

q

, 2

≤ 2 log

N. Applying (9), (13), (14), (54) and (55) we obtain

A(γ, N, {αq

}

) =

k−1

X

m=1

A(γ, n

, 2

q

, {αq

}

) + A(γ, n

, R, {αq

}

)

=

k−1

X

m=1

(γ2

q

+ O(2

)) + γR + O(2

)

= γN + O(2

) = γN + O(log

N ).

Thus, by (1), the theorem is proved.

Acknowledgments. I am very grateful to Professor Meir Smorodinsky for his hospitality, and to the referee for his valuable suggestions.

References

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School of Mathematical Sciences Tel-Aviv University

69978 Tel-Aviv, Israel E-mail: mlevin@math.tau.ac.il

Received on 30.12.1996

and in revised form on 29.5.1998 (3105)