Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory

Philipp, Walter

doi:10.1090/S0002-9947-1994-1249469-5

Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory
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Trans. Amer. Math. Soc. 345 (1994), 705-727 Request permission

Abstract:

Let $\mathcal {T} = \{ {q_1}, \ldots ,{q_\tau }\}$ be a finite set of coprime integers and let $\{ {n_1},{n_2}, \ldots \}$ denote the mutiplicative semigroup generated by $\mathcal {T}$, and arranged in increasing order. Let ${D_N}(\omega )$ denote the discrepancy of the sequence $\{ {n_k}\omega \} _{k = 1}^N\bmod 1$, $\omega \in [0,1)$. In this paper we solve a problem posed by R.C. Baker [3], by proving that for all $\omega$ except on a set of Lebesgue measure 0 \[ \frac {1}{4} \leq \lim \sup \limits _{N \to \infty } \frac {{N{D_N}(\omega )}}{{\sqrt {N\log \log N} }} \leq C.\] Here the constant C only depends on the total number of primes involved in the prime factorization of ${q_1}, \ldots ,{q_\tau }$. The lower bound is obtained from a strong approximation theorem for the partial sums of the sequence $\{ \cos 2\pi {n_k}\omega \} _{k = 1}^\infty$ by sums of independent standard normal random variables.

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