A metric discrepancy result with given speed

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• 作者：I. Berkes ; K. Fukuyama ; T. Nishimura
• 关键词：and phraseslacunary sequence ; discrepancy ; law of the iterated logarithm
• 刊名：Acta Mathematica Hungarica
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：151
• 期：1
• 页码：199-216
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Netherlands
• ISSN：1588-2632
• 卷排序：151

文摘

It is known that the discrepancy \({D_N\{kx\}}\) of the sequence \({\{kx\}}\) satisfies \({ND_N\{kx\} = O((\log N){(\log \log N)}^{1+\varepsilon})}\) a.e. for all \({\varepsilon > 0}\), but not for \({\varepsilon=0}\). For \({n_k=\theta^k}\), \({\theta > 1}\) we have \({ND_N\{n_kx\} \leqq (\Sigma_\theta +\varepsilon){(2N\log \log N)}^{1/2}}\) a.e. for some \({0 < \Sigma_\theta < \infty}\) and \({N\geqq N_0}\) if \({\varepsilon > 0}\), but not for \({\varepsilon < 0}\). In this paper we prove, extending results of Aistleitner–Larcher [6], that for any sufficiently smooth intermediate speed \({\Psi(N)}\) between \({(\log N){(\log \log N)}^{1+\varepsilon}}\) and \({{(N\log \log N)}^{1/2}}\) and for any \({\Sigma > 0}\), there exists a sequence \({\{n_k\}}\) of positive integers such that \({ND_N\{n_kx\} \leqq (\Sigma+\varepsilon)\Psi(N)}\) eventually holds a.e. for \({\varepsilon > 0}\), but not for \({\varepsilon < 0}\). We also consider a similar problem on the growth of trigonometric sums.