{"title":"圆和球面的内禀丢番图近似","authors":"Byungchul Cha, Dong Han Kim","doi":"10.1112/mtk.12228","DOIUrl":null,"url":null,"abstract":"<p>We study Lagrange spectra arising from intrinsic Diophantine approximation of circles and spheres. More precisely, we consider three circles embedded in <math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mn>2</mn>\n </msup>\n <annotation>$\\mathbb {R}^2$</annotation>\n </semantics></math> or <math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mn>3</mn>\n </msup>\n <annotation>$\\mathbb {R}^3$</annotation>\n </semantics></math> and three spheres embedded in <math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mn>3</mn>\n </msup>\n <annotation>$\\mathbb {R}^3$</annotation>\n </semantics></math> or <math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mn>4</mn>\n </msup>\n <annotation>$\\mathbb {R}^4$</annotation>\n </semantics></math>. We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of <math>\n <semantics>\n <mi>R</mi>\n <annotation>$\\mathbb {R}$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mi>C</mi>\n <annotation>$\\mathbb {C}$</annotation>\n </semantics></math>. Thanks to prior work of Asmus L. Schmidt on the spectra of <math>\n <semantics>\n <mi>R</mi>\n <annotation>$\\mathbb {R}$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mi>C</mi>\n <annotation>$\\mathbb {C}$</annotation>\n </semantics></math>, we obtain as a corollary, for each of the six spectra, the smallest accumulation point and the initial discrete part leading up to it completely.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Intrinsic Diophantine approximation on circles and spheres\",\"authors\":\"Byungchul Cha, Dong Han Kim\",\"doi\":\"10.1112/mtk.12228\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study Lagrange spectra arising from intrinsic Diophantine approximation of circles and spheres. More precisely, we consider three circles embedded in <math>\\n <semantics>\\n <msup>\\n <mi>R</mi>\\n <mn>2</mn>\\n </msup>\\n <annotation>$\\\\mathbb {R}^2$</annotation>\\n </semantics></math> or <math>\\n <semantics>\\n <msup>\\n <mi>R</mi>\\n <mn>3</mn>\\n </msup>\\n <annotation>$\\\\mathbb {R}^3$</annotation>\\n </semantics></math> and three spheres embedded in <math>\\n <semantics>\\n <msup>\\n <mi>R</mi>\\n <mn>3</mn>\\n </msup>\\n <annotation>$\\\\mathbb {R}^3$</annotation>\\n </semantics></math> or <math>\\n <semantics>\\n <msup>\\n <mi>R</mi>\\n <mn>4</mn>\\n </msup>\\n <annotation>$\\\\mathbb {R}^4$</annotation>\\n </semantics></math>. We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of <math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$\\\\mathbb {R}$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mi>C</mi>\\n <annotation>$\\\\mathbb {C}$</annotation>\\n </semantics></math>. Thanks to prior work of Asmus L. Schmidt on the spectra of <math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$\\\\mathbb {R}$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mi>C</mi>\\n <annotation>$\\\\mathbb {C}$</annotation>\\n </semantics></math>, we obtain as a corollary, for each of the six spectra, the smallest accumulation point and the initial discrete part leading up to it completely.</p>\",\"PeriodicalId\":18463,\"journal\":{\"name\":\"Mathematika\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-10-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematika\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12228\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12228","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Intrinsic Diophantine approximation on circles and spheres
We study Lagrange spectra arising from intrinsic Diophantine approximation of circles and spheres. More precisely, we consider three circles embedded in or and three spheres embedded in or . We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of and . Thanks to prior work of Asmus L. Schmidt on the spectra of and , we obtain as a corollary, for each of the six spectra, the smallest accumulation point and the initial discrete part leading up to it completely.
期刊介绍：
Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.
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