Transactions of the American Mathematical Society

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


Author: Walter Philipp
Journal: Trans. Amer. Math. Soc. 345 (1994), 705-727
MSC: Primary 11K31; Secondary 11K06, 60F15
MathSciNet review: 1249469
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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

$\displaystyle \frac{1}{4} \leq \mathop {\lim \sup }\limits_{N \to \infty } \frac{{N{D_N}(\omega )}}{{\sqrt {N\log \log N} }} \leq C.$

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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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1249469-5
Article copyright: © Copyright 1994 American Mathematical Society