## Abstract

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.

Original language | English |
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Pages (from-to) | 3259-3280 |

Number of pages | 22 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 May 2019 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics