{"title":"$\\{n\\ α \\}$-序列局部差的单侧有界性","authors":"J. Ying, Yushu Zheng","doi":"10.4064/aa211015-12-11","DOIUrl":null,"url":null,"abstract":"The main interest of this article is the one-sided boundedness of the local discrepancy of $\\alpha\\in\\mathbb{R}\\setminus\\mathbb{Q}$ on the interval $(0,c)\\subset(0,1)$ defined by \\[D_n(\\alpha,c)=\\sum_{j=1}^n 1_{\\{\\{j\\alpha\\}<c\\}}-cn.\\] We focus on the special case $c\\in (0,1)\\cap\\mathbb{Q}$. Several necessary and sufficient conditions on $\\alpha$ for $(D_n(\\alpha,c))$ to be one-side bounded are derived. Using these, certain topological properties are given to describe the size of the set \\[O_c=\\{\\alpha\\in \\irr: (D_n(\\alpha,c)) \\text{ is one-side bounded}\\}.\\]","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2022-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the one-sided boundedness of the local discrepancy of $\\\\{n\\\\alpha \\\\}$-sequences\",\"authors\":\"J. Ying, Yushu Zheng\",\"doi\":\"10.4064/aa211015-12-11\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The main interest of this article is the one-sided boundedness of the local discrepancy of $\\\\alpha\\\\in\\\\mathbb{R}\\\\setminus\\\\mathbb{Q}$ on the interval $(0,c)\\\\subset(0,1)$ defined by \\\\[D_n(\\\\alpha,c)=\\\\sum_{j=1}^n 1_{\\\\{\\\\{j\\\\alpha\\\\}<c\\\\}}-cn.\\\\] We focus on the special case $c\\\\in (0,1)\\\\cap\\\\mathbb{Q}$. Several necessary and sufficient conditions on $\\\\alpha$ for $(D_n(\\\\alpha,c))$ to be one-side bounded are derived. Using these, certain topological properties are given to describe the size of the set \\\\[O_c=\\\\{\\\\alpha\\\\in \\\\irr: (D_n(\\\\alpha,c)) \\\\text{ is one-side bounded}\\\\}.\\\\]\",\"PeriodicalId\":37888,\"journal\":{\"name\":\"Acta Arithmetica\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-10-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Arithmetica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4064/aa211015-12-11\",\"RegionNum\":3,\"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":"Acta Arithmetica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4064/aa211015-12-11","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the one-sided boundedness of the local discrepancy of $\{n\alpha \}$-sequences
The main interest of this article is the one-sided boundedness of the local discrepancy of $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ on the interval $(0,c)\subset(0,1)$ defined by \[D_n(\alpha,c)=\sum_{j=1}^n 1_{\{\{j\alpha\}
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