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.References
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Additional Information
- István Berkes
- Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1053 Budapest, Reáltanoda u. 13-15, Hungary
- MR Author ID: 35400
- Email: berkes.istvan@renyi.mta.hu
- Bence Borda
- Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1053 Budapest, Reáltanoda u. 13-15, Hungary
- MR Author ID: 1186965
- Email: borda.bence@renyi.mta.hu
- Received by editor(s): May 7, 2017
- Received by editor(s) in revised form: August 17, 2017
- Published electronically: December 7, 2018
- Additional Notes: The first author’s research was supported by FWF Grant P24302-N18 and NKFIH grant K 125569.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 3259-3280
- MSC (2010): Primary 42A55; Secondary 42A61, 30B50, 11K38
- DOI: https://doi.org/10.1090/tran/7415
- MathSciNet review: 3896111