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.