{"title":"An optimal lower bound for the discrepancy of c-uniformly distributed functions modulo","authors":"Michael Drmota","doi":"10.1016/1385-7258(88)90004-2","DOIUrl":null,"url":null,"abstract":"<div><p>It is proved that a lower bound for the discrepancy <em>D</em><sub><em>T</em></sub>(<em>ω</em>) of a continuous function <em>ω</em>(<em>t</em>) modulo 1 is given by <em>ϕ</em>(<em>s</em>(<em>T</em>)) infinitely often, where <em>ϕ</em>(<em>x</em>) is integrable on [0, ∞). This bound is best possible in the one dimensional case. Furthermore a generalization to compact, metric spaces is given.</p></div>","PeriodicalId":100664,"journal":{"name":"Indagationes Mathematicae (Proceedings)","volume":"91 1","pages":"Pages 21-28"},"PeriodicalIF":0.0000,"publicationDate":"1988-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/1385-7258(88)90004-2","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae (Proceedings)","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/1385725888900042","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2004/11/15 0:00:00","PubModel":"Epub","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
It is proved that a lower bound for the discrepancy DT(ω) of a continuous function ω(t) modulo 1 is given by ϕ(s(T)) infinitely often, where ϕ(x) is integrable on [0, ∞). This bound is best possible in the one dimensional case. Furthermore a generalization to compact, metric spaces is given.
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