{"title":"On a criterion of uniform distribution","authors":"Grigori Karagulyan, Iren Petrosyan","doi":"arxiv-2408.07061","DOIUrl":null,"url":null,"abstract":"We give an extension of a criterion of van der Corput on uniform distribution\nof sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed\nmodulo 1 if it is weakly monotonic and satisfies the conditions $\\Delta^2x_n\\to\n0,\\quad n^2\\Delta^2x_n\\to \\infty $. Our proof is straightforward and uses a\nDiophantine approximation by rational numbers, while van der Corput's approach\nis based on some estimates of exponential sums.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"64 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Classical Analysis and ODEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.07061","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We give an extension of a criterion of van der Corput on uniform distribution
of sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed
modulo 1 if it is weakly monotonic and satisfies the conditions $\Delta^2x_n\to
0,\quad n^2\Delta^2x_n\to \infty $. Our proof is straightforward and uses a
Diophantine approximation by rational numbers, while van der Corput's approach
is based on some estimates of exponential sums.
我们给出了 van der Corput 关于序列均匀分布准则的扩展。也就是说,我们证明,如果一个序列 $x_n$ 是弱单调的,并且满足条件 $\Delta^2x_n\to0,\quad n^2\Delta^2x_n\to \infty $,那么这个序列就是均匀分布的。 我们的证明简单明了,使用的是有理数的二叉近似,而 van der Corput 的方法是基于指数和的一些估计。