{"title":"一维准均匀克罗内克序列","authors":"Takashi Goda","doi":"10.1007/s00013-024-02039-0","DOIUrl":null,"url":null,"abstract":"<div><p>In this short note, we prove that the one-dimensional Kronecker sequence <span>\\(i\\alpha \\bmod 1, i=0,1,2,\\ldots ,\\)</span> is quasi-uniform over the unit interval [0, 1] if and only if <span>\\(\\alpha \\)</span> is a badly approximable number. Our elementary proof relies on a result on the three-gap theorem for Kronecker sequences due to Halton (Proc Camb Philos Soc, 61:665–670, 1965).</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 5","pages":"499 - 505"},"PeriodicalIF":0.5000,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02039-0.pdf","citationCount":"0","resultStr":"{\"title\":\"One-dimensional quasi-uniform Kronecker sequences\",\"authors\":\"Takashi Goda\",\"doi\":\"10.1007/s00013-024-02039-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this short note, we prove that the one-dimensional Kronecker sequence <span>\\\\(i\\\\alpha \\\\bmod 1, i=0,1,2,\\\\ldots ,\\\\)</span> is quasi-uniform over the unit interval [0, 1] if and only if <span>\\\\(\\\\alpha \\\\)</span> is a badly approximable number. Our elementary proof relies on a result on the three-gap theorem for Kronecker sequences due to Halton (Proc Camb Philos Soc, 61:665–670, 1965).</p></div>\",\"PeriodicalId\":8346,\"journal\":{\"name\":\"Archiv der Mathematik\",\"volume\":\"123 5\",\"pages\":\"499 - 505\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-08-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00013-024-02039-0.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Archiv der Mathematik\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00013-024-02039-0\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archiv der Mathematik","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00013-024-02039-0","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
In this short note, we prove that the one-dimensional Kronecker sequence \(i\alpha \bmod 1, i=0,1,2,\ldots ,\) is quasi-uniform over the unit interval [0, 1] if and only if \(\alpha \) is a badly approximable number. Our elementary proof relies on a result on the three-gap theorem for Kronecker sequences due to Halton (Proc Camb Philos Soc, 61:665–670, 1965).
期刊介绍：
Archiv der Mathematik (AdM) publishes short high quality research papers in every area of mathematics which are not overly technical in nature and addressed to a broad readership.
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