{"title":"关于薄扇形的晶格点数","authors":"Ezra Waxman, Nadav Yesha","doi":"10.1007/s00605-024-01983-x","DOIUrl":null,"url":null,"abstract":"<p>On the circle of radius <i>R</i> centred at the origin, consider a “thin” sector about the fixed line <span>\\(y = \\alpha x\\)</span> with edges given by the lines <span>\\(y = (\\alpha \\pm \\epsilon ) x\\)</span>, where <span>\\(\\epsilon = \\epsilon _R \\rightarrow 0\\)</span> as <span>\\( R \\rightarrow \\infty \\)</span>. We establish an asymptotic count for <span>\\(S_{\\alpha }(\\epsilon ,R)\\)</span>, the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of <span>\\(\\epsilon \\)</span> and on the rationality/irrationality type of <span>\\(\\alpha \\)</span>. In particular, we demonstrate that if <span>\\(\\alpha \\)</span> is Diophantine, then <span>\\(S_{\\alpha }(\\epsilon ,R)\\)</span> is asymptotic to the area of the sector, so long as <span>\\(\\epsilon R^{t} \\rightarrow \\infty \\)</span> for some <span>\\( t<2 \\)</span>.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the number of lattice points in thin sectors\",\"authors\":\"Ezra Waxman, Nadav Yesha\",\"doi\":\"10.1007/s00605-024-01983-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>On the circle of radius <i>R</i> centred at the origin, consider a “thin” sector about the fixed line <span>\\\\(y = \\\\alpha x\\\\)</span> with edges given by the lines <span>\\\\(y = (\\\\alpha \\\\pm \\\\epsilon ) x\\\\)</span>, where <span>\\\\(\\\\epsilon = \\\\epsilon _R \\\\rightarrow 0\\\\)</span> as <span>\\\\( R \\\\rightarrow \\\\infty \\\\)</span>. We establish an asymptotic count for <span>\\\\(S_{\\\\alpha }(\\\\epsilon ,R)\\\\)</span>, the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of <span>\\\\(\\\\epsilon \\\\)</span> and on the rationality/irrationality type of <span>\\\\(\\\\alpha \\\\)</span>. In particular, we demonstrate that if <span>\\\\(\\\\alpha \\\\)</span> is Diophantine, then <span>\\\\(S_{\\\\alpha }(\\\\epsilon ,R)\\\\)</span> is asymptotic to the area of the sector, so long as <span>\\\\(\\\\epsilon R^{t} \\\\rightarrow \\\\infty \\\\)</span> for some <span>\\\\( t<2 \\\\)</span>.</p>\",\"PeriodicalId\":18913,\"journal\":{\"name\":\"Monatshefte für Mathematik\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Monatshefte für Mathematik\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00605-024-01983-x\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-01983-x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the circle of radius R centred at the origin, consider a “thin” sector about the fixed line \(y = \alpha x\) with edges given by the lines \(y = (\alpha \pm \epsilon ) x\), where \(\epsilon = \epsilon _R \rightarrow 0\) as \( R \rightarrow \infty \). We establish an asymptotic count for \(S_{\alpha }(\epsilon ,R)\), the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of \(\epsilon \) and on the rationality/irrationality type of \(\alpha \). In particular, we demonstrate that if \(\alpha \) is Diophantine, then \(S_{\alpha }(\epsilon ,R)\) is asymptotic to the area of the sector, so long as \(\epsilon R^{t} \rightarrow \infty \) for some \( t<2 \).