A metric discrepancy result with given speed

I. Berkes, K. Fukuyama*, T. Nishimura

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

It is known that the discrepancy DN{ kx} of the sequence { kx} satisfies NDN{ kx} = O((log N) (log log N) 1 + ε) a.e. for all ε> 0 , but not for ε= 0. For nk= θk, θ> 1 we have NDN{ nkx} ≦ (Σ θ+ ε) (2 Nlog log N) 1 / 2 a.e. for some 0 < Σ θ< ∞ and N≧ N0 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 { nk} of positive integers such that NDN{ nkx} ≦ (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 languageEnglish
Pages (from-to)199-216
Number of pages18
JournalActa Mathematica Hungarica
Volume151
Issue number1
DOIs
Publication statusPublished - 1 Feb 2017

Keywords

  • discrepancy
  • lacunary sequence
  • law of the iterated logarithm

ASJC Scopus subject areas

  • Mathematics(all)

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