Hauptmenü
  • Autor
    • Berkes, István
    • Fukuyama, Katusi
    • Nishimura, Takuya
  • TitelA Metric Discrepancy Result With Given Speed
  • Volltext
  • DOI10.1007/s10474-016-0658-2
  • Persistent Identifier
  • Erschienen inActa Mathematica Hungarica
  • Band151
  • Erscheinungsjahr2017
  • Heft1
  • Seiten199-216
  • LicenceCC BY
  • ISSN0236-5294
  • ZugriffsrechteCC-BY
  • Download Statistik4
  • Peer ReviewJa
  • AbstractIt is known that the discrepancy \(D_N\{kx\}\) quence \(\{kx\}\) 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 \(ND_N \{n_kx\} ≤ (Σ_{θ + ε})(2N log log N )^{1/2}\) 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 \((N log log N )^{1/2}\) and for any \(Σ > 0\), there exists a sequence \(\{n_k\}\) of positive integers such that \(N D_N \{n_k x\} ≤ (Σ + ε)Ψ(N)\) eventually holds a.e. for \(ε > 0\), but not for \(ε < 0\). We also consider a similar problem on the growth of trigonometric sums.