On the law of the iterated logarithm for random exponential sums

The asymptotic behavior of exponential sums ^N _k=1 exp(2πin _k α) for Hadamard lacunary (n _k ) is well known, but for general (n _k ) very few precise results exist, due to number theoretic difficulties. It is therefore natural to consider random∑ (n _k ), and in this paper we prove the law of the iterated logarithm for ^N _k=1 exp(2πin _k α) if the gaps n _k+1 −n _k are independent, identically distributed random variables. As a comparison, we give a lower bound for the discrepancy of {n _k α} under the same random model, exhibiting a completely different behavior.