Empirical processes in probabilistic number theory: The LIL for the discrepancy of (nkω) MOD 1

István Berkes*, Walter Philipp, Robert F. Tichy

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We prove a law of the iterated logarithm for the Kolmogorov-Smirnov statistic, or equivalently, the discrepancy of sequences (nkω) mod 1. Here (nk) is a sequence of integers satisfying a sub-Hadamard growth condition and such that linear Diophantine equations in the variables nk 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.

Original languageEnglish
Pages (from-to)107-145
Number of pages39
JournalIllinois journal of mathematics
Volume50
Issue number1
DOIs
Publication statusPublished - 1 Jan 2006

ASJC Scopus subject areas

  • Mathematics(all)

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