### 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 |
---|---|

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 - May 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*371*(5), 3259-3280. https://doi.org/10.1090/tran/7415