A metric discrepancy result with given speed

I. Berkes, K. Fukuyama, T. Nishimura

Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

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.

Originalspracheenglisch
Seiten (von - bis)199-216
Seitenumfang18
FachzeitschriftActa Mathematica Hungarica
Jahrgang151
Ausgabenummer1
DOIs
PublikationsstatusVeröffentlicht - 1 Feb 2017

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Discrepancy
Trigonometric Sums
Metric
Integer

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    A metric discrepancy result with given speed. / Berkes, I.; Fukuyama, K.; Nishimura, T.

    in: Acta Mathematica Hungarica, Jahrgang 151, Nr. 1, 01.02.2017, S. 199-216.

    Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

    Berkes, I. ; Fukuyama, K. ; Nishimura, T. / A metric discrepancy result with given speed. in: Acta Mathematica Hungarica. 2017 ; Jahrgang 151, Nr. 1. S. 199-216.
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