On the law of the iterated logarithm for the discrepancy of lacunary sequences

Aistleitner, Christoph

doi:10.1090/S0002-9947-2010-05026-3

On the law of the iterated logarithm for the discrepancy of lacunary sequences
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by Christoph Aistleitner

Trans. Amer. Math. Soc. 362 (2010), 5967-5982

DOI: https://doi.org/10.1090/S0002-9947-2010-05026-3

Published electronically: June 16, 2010

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Abstract:

A classical result of Philipp (1975) states that for any sequence $(n_k)_{k \geq 1}$ of integers satisfying the Hadamard gap condition $n_{k+1}/n_k\ge q>1 \ (k=1, 2, \ldots )$, the discrepancy $D_N$ of the sequence $(n_k x)_{k\ge 1}$ mod 1 satisfies the law of the iterated logarithm (LIL), i.e. \[ 1/4 \leq \limsup _{N \to \infty } N D_N(n_k x) (N \log \log N)^{-1/2} \leq C_q \quad \mathrm {a.e.}\] The value of the $\limsup$ is a long-standing open problem. Recently Fukuyama explicitly calculated the value of the $\limsup$ for $n_k= \theta ^k$, $\theta >1$, not necessarily integer. We extend Fukuyama’s result to a large class of integer sequences $(n_k)$ characterized in terms of the number of solutions of a certain class of Diophantine equations and show that the value of the $\limsup$ is the same as in the Chung-Smirnov LIL for i.i.d. random variables.

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