{"title":"关于同时情况下精确可近似向量集的评论","authors":"Reynold Fregoli","doi":"10.1090/proc/16790","DOIUrl":null,"url":null,"abstract":"<p>We compute the Hausdorff dimension of the set of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"psi\">\n <mml:semantics>\n <mml:mi>ψ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\psi</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-exactly approximable vectors, in the simultaneous case, in dimension strictly larger than <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\n <mml:semantics>\n <mml:mn>2</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and for approximating functions <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"psi\">\n <mml:semantics>\n <mml:mi>ψ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\psi</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with order at infinity less than or equal to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"negative 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo>−</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">-2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Our method relies on the analogous result in dimension <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\">\n <mml:semantics>\n <mml:mn>1</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník’s theorem on fibres.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A remark on the set of exactly approximable vectors in the simultaneous case\",\"authors\":\"Reynold Fregoli\",\"doi\":\"10.1090/proc/16790\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We compute the Hausdorff dimension of the set of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"psi\\\">\\n <mml:semantics>\\n <mml:mi>ψ</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\psi</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-exactly approximable vectors, in the simultaneous case, in dimension strictly larger than <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"2\\\">\\n <mml:semantics>\\n <mml:mn>2</mml:mn>\\n <mml:annotation encoding=\\\"application/x-tex\\\">2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and for approximating functions <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"psi\\\">\\n <mml:semantics>\\n <mml:mi>ψ</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\psi</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> with order at infinity less than or equal to <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"negative 2\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo>−</mml:mo>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">-2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Our method relies on the analogous result in dimension <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"1\\\">\\n <mml:semantics>\\n <mml:mn>1</mml:mn>\\n <mml:annotation encoding=\\\"application/x-tex\\\">1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník’s theorem on fibres.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-06-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16790\",\"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":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16790","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
A remark on the set of exactly approximable vectors in the simultaneous case
We compute the Hausdorff dimension of the set of ψ\psi-exactly approximable vectors, in the simultaneous case, in dimension strictly larger than 22 and for approximating functions ψ\psi with order at infinity less than or equal to −2-2. Our method relies on the analogous result in dimension 11, proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník’s theorem on fibres.
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