{"title":"Diophantine approximation 中 limsup 集笛卡尔积的 Hausdorff 维度","authors":"Baowei Wang, Jun Wu","doi":"10.1090/tran/9136","DOIUrl":null,"url":null,"abstract":"<p>The metric theory of limsup sets is the main topic in metric Diophantine approximation. A very simple observation by Erdös shows the dimension of the Cartesian product of two sets of Liouville numbers is 1. To disclose the mystery hidden there, we consider and present a general principle for the Hausdorff dimension of the Cartesian product of limsup sets. As an application of our general principle, it is found that <disp-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"dimension Subscript script upper H Baseline upper W left-parenthesis psi right-parenthesis times midline-horizontal-ellipsis times upper W left-parenthesis psi right-parenthesis equals d minus 1 plus dimension Subscript script upper H Baseline upper W left-parenthesis psi right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>dim</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"script\">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo><!-- --></mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>×<!-- × --></mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>×<!-- × --></mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>dim</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"script\">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo><!-- --></mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\begin{equation*} \\dim _{\\mathcal H}W(\\psi )\\times \\cdots \\times W(\\psi )=d-1+\\dim _{\\mathcal H}W(\\psi ) \\end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper W left-parenthesis psi right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">W(\\psi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the set of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"psi\"> <mml:semantics> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-well approximable points in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"psi colon double-struck upper N right-arrow double-struck upper R Superscript plus\"> <mml:semantics> <mml:mrow> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">→<!-- → --></mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\psi : \\mathbb {N}\\to \\mathbb {R}^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a positive non-increasing function. Even this concrete case was never observed before. Our result can also be compared with Marstrand’s famous inequality on the dimension of the Cartesian product of general sets.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":"33 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Hausdorff dimension of the Cartesian product of limsup sets in Diophantine approximation\",\"authors\":\"Baowei Wang, Jun Wu\",\"doi\":\"10.1090/tran/9136\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The metric theory of limsup sets is the main topic in metric Diophantine approximation. A very simple observation by Erdös shows the dimension of the Cartesian product of two sets of Liouville numbers is 1. To disclose the mystery hidden there, we consider and present a general principle for the Hausdorff dimension of the Cartesian product of limsup sets. As an application of our general principle, it is found that <disp-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"dimension Subscript script upper H Baseline upper W left-parenthesis psi right-parenthesis times midline-horizontal-ellipsis times upper W left-parenthesis psi right-parenthesis equals d minus 1 plus dimension Subscript script upper H Baseline upper W left-parenthesis psi right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>dim</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"script\\\">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo><!-- --></mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>×<!-- × --></mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>×<!-- × --></mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>dim</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"script\\\">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo><!-- --></mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\begin{equation*} \\\\dim _{\\\\mathcal H}W(\\\\psi )\\\\times \\\\cdots \\\\times W(\\\\psi )=d-1+\\\\dim _{\\\\mathcal H}W(\\\\psi ) \\\\end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper W left-parenthesis psi right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">W(\\\\psi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the set of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"psi\\\"> <mml:semantics> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-well approximable points in <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R\\\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"psi colon double-struck upper N right-arrow double-struck upper R Superscript plus\\\"> <mml:semantics> <mml:mrow> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">N</mml:mi> </mml:mrow> <mml:mo stretchy=\\\"false\\\">→<!-- → --></mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\psi : \\\\mathbb {N}\\\\to \\\\mathbb {R}^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a positive non-increasing function. Even this concrete case was never observed before. Our result can also be compared with Marstrand’s famous inequality on the dimension of the Cartesian product of general sets.</p>\",\"PeriodicalId\":23209,\"journal\":{\"name\":\"Transactions of the American Mathematical Society\",\"volume\":\"33 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-02-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/tran/9136\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9136","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
极限集的度量理论是度量 Diophantine 近似的主要课题。厄尔多斯的一个非常简单的观察结果表明,两个刘维尔数集的笛卡尔积的维数是 1。为了揭示其中隐藏的奥秘,我们考虑并提出了关于跛上集的笛卡尔积的豪斯多夫维数的一般原理。作为我们一般原理的应用,我们发现 dim H W ( ψ ) × ⋯ × W ( ψ ) = d - 1 + dim H W ( ψ ) (begin{equation*})。\dim _{\mathcal H}W(\psi )\times \cdots \times W(\psi )=d-1+\dim _{\mathcal H}W(\psi ) \end{equation*} 其中 W ( ψ ) W(\psi ) 是 R \mathbb {R} 和 ψ 中 ψ \psi -well approximable points 的集合: N → R + \psi :\到 \mathbb {R}^+ 是一个正的非递增函数。即使是这种具体情况,以前也从未观察到过。我们的结果还可以与马斯特兰关于一般集合的笛卡尔积的维数的著名不等式相比较。
Hausdorff dimension of the Cartesian product of limsup sets in Diophantine approximation
The metric theory of limsup sets is the main topic in metric Diophantine approximation. A very simple observation by Erdös shows the dimension of the Cartesian product of two sets of Liouville numbers is 1. To disclose the mystery hidden there, we consider and present a general principle for the Hausdorff dimension of the Cartesian product of limsup sets. As an application of our general principle, it is found that dimHW(ψ)×⋯×W(ψ)=d−1+dimHW(ψ)\begin{equation*} \dim _{\mathcal H}W(\psi )\times \cdots \times W(\psi )=d-1+\dim _{\mathcal H}W(\psi ) \end{equation*} where W(ψ)W(\psi ) is the set of ψ\psi-well approximable points in R\mathbb {R} and ψ:N→R+\psi : \mathbb {N}\to \mathbb {R}^+ is a positive non-increasing function. Even this concrete case was never observed before. Our result can also be compared with Marstrand’s famous inequality on the dimension of the Cartesian product of general sets.
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