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2006 Empirical processes in probabilistic number theory: the LIL for the discrepancy of $(n\sb k\omega)\bmod1$
István Berkes, Walter Philipp, Robert F. Tichy
Illinois J. Math. 50(1-4): 107-145 (2006). DOI: 10.1215/ijm/1258059472

Abstract

We prove a law of the iterated logarithm for the Kolmogorov-Smirnov statistic, or equivalently, the discrepancy of sequences $(n_{k}\omega)$ mod $1$. Here $(n_{k})$ is a sequence of integers satisfying a sub-Hadamard growth condition and such that linear Diophantine equations in the variables $n_{k}$ do not have too many solutions. The proof depends on a martingale embedding of the empirical process; the number-theoretic structure of $(n_k)$ enters through the behavior of the square function of the martingale.

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István Berkes. Walter Philipp. Robert F. Tichy. "Empirical processes in probabilistic number theory: the LIL for the discrepancy of $(n\sb k\omega)\bmod1$." Illinois J. Math. 50 (1-4) 107 - 145, 2006. https://doi.org/10.1215/ijm/1258059472

Information

Published: 2006
First available in Project Euclid: 12 November 2009

zbMATH: 1145.11058
MathSciNet: MR2247826
Digital Object Identifier: 10.1215/ijm/1258059472

Subjects:
Primary: 60F15
Secondary: 11K06, 11K38

Rights: Copyright © 2006 University of Illinois at Urbana-Champaign

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Vol.50 • No. 1-4 • 2006
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