EMPIRICAL PROCESSES IN PROBABILISTIC NUMBER THEORY: THE LIL FOR THE DISCREPANCY OF (n_kw) MOD 1

摘要

We prove a law of the iterated logarithm for the Kolmogorov-Smirnov statistic, or equivalently, the discrepancy of sequences (n_kw) 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.