Empirical processes in probabilistic number theory: the LIL for the discrepancy of \((n_{k}\omega)\bmod 1\) (Q2505467)

Let \((n_k)\) be an increasing sequence of positive integers satisfying a sub-Hadamard growth condition and certain Diophantine properties. The authors prove a law of the iterated logarithm for the discrepancy \(D_N(\omega )\) of the sequence \(\{n_k\omega\}\). That is, there is a constant \(D\) such that \[ \frac14\leq \limsup_{N\to \infty} \frac{ND_N(\omega)}{\sqrt{N\log \log N}} \leq D \] for almost all \(\omega\in[0,1)\). Furthermore, for the empirical distribution function \(F_n(t)\) of the sequence \(\{n_k\omega\}\) the authors give the following stronger theorem. There exist constants \(\delta>0\) and \(D\) such that for all \(N\geq N_0\) and all \(s\) and \(t\) with \(0\leq s<t\leq 1\) \[ \max_{n\leq N} n| F_n(t)-F_n(s)-(t-s)| \leq D(t-s)^{\delta}(N\log\log N) ^{1/2} +N^{1/2} \] for almost all \(\omega\in [0,1)\).