On a criterion of uniform distribution

Grigori Karagulyan, Iren Petrosyan
{"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.
关于均匀分布的标准
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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信