On the law of the iterated logarithm for random exponential sums

The asymptotic behavior of exponential sums ∑ k = 1 N exp ⁡ ( 2 π i n k α ) \sum _{k=1}^N \exp ( 2\pi i n_k \alpha ) for Hadamard lacunary ( n k ) (n_k) is well known, but for general ( n k ) (n_k) very few precise results exist, due to number theoretic difficulties. It is therefore natural to consider random ( n k ) (n_k) , and in this paper we prove the law of the iterated logarithm for ∑ k = 1 N exp ⁡ ( 2 π i n k α ) \sum _{k=1}^N \exp (2\pi i n_k \alpha ) if the gaps n k + 1 − n k 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 } \{n_k \alpha \} under the same random model, exhibiting a completely different behavior.