- Autor
- Berkes, István
- Fukuyama, Katusi
- Nishimura, Takuya
- TitelA Metric Discrepancy Result With Given Speed
- Datei
- DOI10.1007/s10474-016-0658-2
- Persistent Identifier
- Erschienen inActa Mathematica Hungarica
- Band151
- Erscheinungsjahr2017
- Heft1
- Seiten199-216
- LicenceCC BY
- ISSN0236-5294
- Download Statistik2077
- 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.
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