Empirical processes in probabilistic number theory: the LIL for the discrepancy of $(n\sb k\omega)\bmod1$

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.