{"title":"博雷尔关于二相逼近的一个定理","authors":"Jaroslav Hančl, Radhakrishnan Nair","doi":"10.1007/s11139-024-00922-6","DOIUrl":null,"url":null,"abstract":"<p>A theorem of É. Borel’s asserts that one of any three consecutive convergents of a real number <i>a</i>, which we denote <span>\\(\\frac{p}{q}\\)</span>, satisfies the inequality <span>\\(\\left| a-\\frac{p}{q} \\right| < \\frac{C}{q^2}\\)</span> with <span>\\(C=\\frac{1}{\\sqrt{5}}\\)</span>. In this paper we give more precise information about the constant <i>C</i>.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"25 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On a theorem of Borel on diophantine approximation\",\"authors\":\"Jaroslav Hančl, Radhakrishnan Nair\",\"doi\":\"10.1007/s11139-024-00922-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A theorem of É. Borel’s asserts that one of any three consecutive convergents of a real number <i>a</i>, which we denote <span>\\\\(\\\\frac{p}{q}\\\\)</span>, satisfies the inequality <span>\\\\(\\\\left| a-\\\\frac{p}{q} \\\\right| < \\\\frac{C}{q^2}\\\\)</span> with <span>\\\\(C=\\\\frac{1}{\\\\sqrt{5}}\\\\)</span>. In this paper we give more precise information about the constant <i>C</i>.</p>\",\"PeriodicalId\":501430,\"journal\":{\"name\":\"The Ramanujan Journal\",\"volume\":\"25 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Ramanujan Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s11139-024-00922-6\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Ramanujan Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11139-024-00922-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
Borel 的一个定理断言Borel's 断言，实数 a 的任意三个连续收敛数中的一个，我们用 \(\frac{p}{q}\) 表示，满足不等式 \(\left| a-\frac{p}{q} \right| < \frac{C}{q^2}\) with \(C=\frac{1}{\sqrt{5}}\).本文将给出关于常数 C 的更精确信息。
On a theorem of Borel on diophantine approximation
A theorem of É. Borel’s asserts that one of any three consecutive convergents of a real number a, which we denote \(\frac{p}{q}\), satisfies the inequality \(\left| a-\frac{p}{q} \right| < \frac{C}{q^2}\) with \(C=\frac{1}{\sqrt{5}}\). In this paper we give more precise information about the constant C.
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