Lobachevskii Journal of Mathematics Vol. 11, 2002, 22-25 \begin{matrix}c \\ ◯ \end{matrix}L. Pushkin

L. Pushkin

SMALL DIGITWISE PERTURBATIONS OF A NUMBER MAKE IT NORMAL TO UNRELATED BASES

(submitted by D. Kh. Mustari)

Abstract. Let r, g ≥ 2 be integers such that log g/ log r is irrational. We show that under r-digitwise random perturbations of an expanded to base r real number x, which are small enough to preserve r-digit asymptotic frequency spectrum of x, the g-adic digits of x tend to have the most chaotic behaviour.

1. Introduction.

Since we are still in the dark about the asymptotic behaviour of digit frequencies of particular irrational numbers such as π, e, \sqrt{2}, log 2, etc., related metric results are of current interest. In this note we generalize a result of Colebrook and Kemperman [3], refined, in turn, the well-known Cassels–Schmidt theorem [2, 6].

We denote by T the additive group of real numbers modulo one. For r ∈ N\{1}, let I_{r} be the set of all probability measures on T that are invariant under the r-adic transformation t → rt mod 1 of T onto itself. For example, the Haar measure on T, call it λ, belongs to I_{r} for every r ∈ N\{1}. For t ∈ T and r ∈ N\{1}, we denote by V_{r}(t) the set of all accumulation points (in the weak^{∗}-topology) of the sequence of empirical measures

\frac{1}{N} ∑_{k=0}^{N−1} δ(tr^{k} mod 1), N ∈ N

on T, where δ(x) is the probability measure concentrated at x. It is known (see [3]) that V_{r}(t) is non-empty closed and connected subset of I_{r}. Recall also, that t ∈ T is said to be normal to the base r ∈ N\{1} if V_{r}(t) = {λ}. It is well-known that λ-almost all t ∈ T are normal to every base r ∈ N\{1}. On the other hand, given any non-empty closed and connected subset J of I_{r} there exists t ∈ T such that V_{r}(t) = J, see [3].

1991 Mathematical Subject Classification. Primary 11K16.

For r, g ∈ N\{1}, we write g ∼ r if the ratio log g/ log r is rational; otherwise g ≁ r. The basic Cassels–Schmidt theorem [2, 6] states that

(A): if g ∼ r then any t ∈ T normal to the base g is normal to the base r; (B): if g ≁ r then the set of numbers t ∈ T which are simultaneously

normal to the base g and non-normal to the base r has the cardinality of the reals.

The following noticeable result is due to Colebrook and Kemperman [3]. For given t ∈ T and r ∈ N\{1}, there exists a number x ∈ T such that

(a): V_{g}(x) = V_{g}(t) for each g ∼ r,

(b): V_{g}(x) = {λ} for each g ≁ r. Theorem. Let t = 0. t_{1}t_{2} … be the canonical r-adic expansion of t ∈ T. Let τ_{1}, τ_{2}, … be statistically independent random r-adic digits, τ = 0. τ_{1}τ_{2} …, and d_{j} = P(τ_{j} ≠ t_{j}).

(A): If d_{j} → 0 as j → ∞, then V_{g}(τ) = V_{g}(t) almost surely for every g ∼ r. (B): If

0 < d_{j} ≤ \frac{1}{2} and \frac{log j}{log log j} d_{j} → ∞ as j → ∞,

then, almost surely, τ is normal to every base g ≁ r.

Remark. If the series ∑ d_{j} diverges while lim d_{j} = 0 then the distribution of τ is non-atomic and singular with respect to λ.

As a consequence we have the following subtle complement to the Cassels– Schmidt theorem which is in essence due to Colebrook and Kemperman [3], cf. [7].

Corollary. For any t ∈ T and r ∈ N\{1}, the set of numbers x ∈ T, such that V_{g}(x) = V_{g}(t) for each g ∼ r while V_{g}(x) = {λ} for each g ≁ r, has the cardinality of the reals.

2. Sketch of the proof of Theorem.

Here we prove the nontrivial part (B). Let Φ^{(τ)}(x) = E exp(iτx), x ∈ R, be the characteristic function of τ, and φ_{j}(x), x ∈ R, be the characteristic functions of τ_{j}, j ∈ N. From the Davenport–Erdös–LeVeque theorem [4] it follows that to prove (B) it suffices to show that random Weyl’s sums

S_{N} = ∑_{k=0}^{N−1} exp(2πiLg^{k}τ), N ∈ N

with fixed L ∈ N satisfy the following condition

\begin{matrix}E|S_{N}|^{2} = ∑_{k′=0}^{N−1} ∑_{k′′=0}^{N−1} Φ^{(τ)}(2πL(g^{k′} − g^{k′′})) = \\ = O (N^{2}/ log^{2} N) as N → ∞. \end{matrix}  (1)

It is easily seen that

E|S_{N}|^{2} ≤ N + 2N max_{h∈H} ∑_{k=0}^{N−1} |Φ^{(τ)} (2πhg^{k})| ,  (2)

where H = {1, 2, …, Lg^{N}}. By the definition of τ

Φ^{(τ)}(x) = ∏_{j=1}^{∞} φ_{j}(x/r^{j}).  (3)

Simple computation shows that

|φ_{j}(2πx)| ≤ 1 − 4r^{−4}d_{j} as x mod 1 ∈ [r^{−2}, r^{−1}) .  (4)

Consider the r-adic expansion

hg^{k} = ε_{1}r^{q−1} + ε_{2}r^{q−2} + ⋯ + ε_{q}, ε_{j} ∈ {0, 1, …, r − 1}, ε_{1} ≠ 0.  (5)

Let K ∈ N, M = 2K+1 and μ^{(h)}(k) be the number of pairs ε_{2j}ε_{2j+1} with ε_{2j} = 0 and ε_{2j+1} ≠ 0 among M leading digits in the r-adic expansion (6); in the case M > q we assume that ε_{q+1} = ⋯ = ε_{M} = 0. Set J_{N} = 2N log_{r} g + log_{r} L and

θ_{N} = 4r^{−4} min_{j≤J_{N}} d_{j}.  (6)

It follows from (4) and (5) that

∑_{k=0}^{N−1} |Φ^{(τ)}(2πhg^{k})| ≤ ∑_{k=0}^{N−1}(1 − θ_{N})^{μ^{(h)}(k)} as h ∈ H.  (7)

Let (δ_{1}, …, δ_{M}) be a block of r-adic digits with δ_{1} ≠ 0. Set

ν^{(h)}(δ, N) = card{k ∈ {0, 1, …, N − 1} | ε_{1} = δ_{1}, …, ε_{M} = δ_{M}}.

Since the condition is satisfied if and only if the fractional part of log_{r}(hg^{k}) falls into the interval

I_{Δ} = [log_{r} ∑_{k=1}^{M} δ_{k}r^{−k+1}, log_{r}(r^{−M+1} + ∑_{k=1}^{M} δ_{k}r^{−k+1})) ,

the theory of the uniform distribution modulo 1 is usable to approximate the value of N^{−1} ν^{(h)}(δ, N) by the length of I_{Δ}.

Lemma. There exist constants D > 0 and ε > 0 depending only on r and g such that

ν^{(h)} (δ, N) ≤ DN^{1− ε} + N/r^{M−1} log r.  (8)

Proof. From Lemma 3.2 Chap. II [5] it follows that there exists an absolute constant C such that for every n ∈ N

Γ_{N} := |N^{−1} ν^{(h)}(δ, N) − log_{r}(1 + 1/ ∑_{k=1}^{M} δ_{k}r^{M−k})| ≤

≤ C( \frac{1}{n} + \frac{1}{N} ∑_{j=1}^{n} \frac{1}{j ∥j log_{r} g∥} ),  (9)

where ∥x∥ is the distance x from the nearest integer. By Feldman’s Theorem (see Baker [1, Theorem 3.1]), g ≁ r yields that ∥j log_{r} g∥ ≥ d j−ω with positive constants d and ω. Choosing suitable n in (10) we obtain the upperbound Γ_{N} ≤ DN −ε with D, ε > 0 as needed for (9). Lemma is proved.

Setting

K = [\frac{ε}{4} log_{r} N]  (10)

we have from (9) that there exists a constant n_{0} depending only on r and g such that

ν^{(h)} (δ, N) ≤ 2 N r^{−2K} as N ≥ n_{0}.  (11)

Rewritting the right-hand side of (8) by summation over all admissible values of μ^{(h)} (k) we have, in view of (12), that

max_{h∈H} ∑_{k=0}^{N−1} |Φ^{(τ)}(2πhg^{k})| ≤

≤ 2Nr^{−2K} ∑_{j=0}^{K} \binom{K}{j}(r − 1)^{j+1}(r^{2} − r + 1)^{K−j} (1 − θ_{N})^{j} =

= 2N(r − 1) (1 − \frac{r − 1}{r^{2}}θ_{N} )^{K} as N ≥ n_{0}.

Now from (7), (11), (1), and (3) we have (2). This completes the proof of Theorem.

References

1. A. Baker. Transcendental number theory. Cambridge University Press, 1975.

2. J. W. S. Cassels. On a problem of Steinhause about normal numbers. Colloquium Math. 7 (1959), 95–101.

3. C. M. Colebrook and J. H. B. Kemperman. On non-normal numbers. Indagationes Math. 30 (1968), 1–11.

4. H. Davenport, P. Erdös and W. J. LeVeque. On Weyl’s criterion for uniform distribution. Michigan Math. J. 10 (1963), 311–314.

5. L. Kuipers and H. Niederreiter. Uniform distribution of sequences. J. Wiley, New York etc., 1974.

6. W. Schmidt. On normal numbers. Pacific J. Math. 10 (1960), 661–672.

7. B. Volkmann. On the Cassels–Schmidt theorem, I. Bull. Sci. Math. 2^{e} série, 108 (1984), 321–336.

Department of Mathematical Statistics, Kazan State University, Kazan 420008, Russia E-mail address: lpushkin@ksu.ru

Received October, 29, 2002