On the law of the iterated logarithm for random exponential sums

Berkes, István and Borda, Bence (2018) On the law of the iterated logarithm for random exponential sums. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. pp. 1-30. ISSN 0002-9947

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Abstract

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

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