Empirical processes in probabilistic number theory : the LIL for the discrepancy of ( n k ω ) mod 1

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 subHadamard 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 (nk) enters through the behavior of the square function of the martingale.