On the law of the iterated logarithm for random exponential sums

Berkes, István; Borda, Bence

doi:10.1090/tran/7415

On the law of the iterated logarithm for random exponential sums
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by István Berkes and Bence Borda PDF

Trans. Amer. Math. Soc. 371 (2019), 3259-3280 Request permission

Abstract:

The asymptotic behavior of exponential sums $\sum _{k=1}^N \exp ( 2\pi i n_k \alpha )$ 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 $\sum _{k=1}^N \exp (2\pi i n_k \alpha )$ 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 $\{n_k \alpha \}$ under the same random model, exhibiting a completely different behavior.

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