## Abstract

It is known that the discrepancy D_{N}{ kx} of the sequence { kx} satisfies ND_{N}{ kx} = O((log N) (log log N) ^{1} ^{+} ^{ε}) a.e. for all ε> 0 , but not for ε= 0. For n_{k}= θ^{k}, θ> 1 we have ND_{N}{ n_{k}x} ≦ (Σ _{θ}+ ε) (2 Nlog log N) ^{1 / 2} a.e. for some 0 < Σ _{θ}< ∞ and N≧ N_{0} if ε> 0 , but not for ε< 0. In this paper we prove, extending results of Aistleitner–Larcher [6], that for any sufficiently smooth intermediate speed Ψ (N) between (log N) (log log N) ^{1} ^{+} ^{ε} and (Nlog log N) ^{1 / 2} and for any Σ > 0 , there exists a sequence { n_{k}} of positive integers such that ND_{N}{ n_{k}x} ≦ (Σ + ε) Ψ (N) eventually holds a.e. for ε> 0 , but not for ε< 0. We also consider a similar problem on the growth of trigonometric sums.

Original language | English |
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Pages (from-to) | 199-216 |

Number of pages | 18 |

Journal | Acta Mathematica Hungarica |

Volume | 151 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Feb 2017 |

## Keywords

- discrepancy
- lacunary sequence
- law of the iterated logarithm

## ASJC Scopus subject areas

- Mathematics(all)