{"title":"线性形式系统的达芬-谢弗猜想","authors":"Felipe A. Ramírez","doi":"10.1112/jlms.12909","DOIUrl":null,"url":null,"abstract":"<p>We extend the Duffin–Schaeffer conjecture to the setting of systems of <span></span><math>\n <semantics>\n <mi>m</mi>\n <annotation>$m$</annotation>\n </semantics></math> linear forms in <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math> variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-by-<span></span><math>\n <semantics>\n <mi>m</mi>\n <annotation>$m$</annotation>\n </semantics></math> systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When <span></span><math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>=</mo>\n <mi>n</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$m=n=1$</annotation>\n </semantics></math>, this is the classical 1941 Duffin–Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher dimensional version, where <span></span><math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>></mo>\n <mn>1</mn>\n </mrow>\n <annotation>$m&gt;1$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$n=1$</annotation>\n </semantics></math>, in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010, Beresnevich and Velani proved the <span></span><math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>></mo>\n <mn>1</mn>\n </mrow>\n <annotation>$m&gt;1$</annotation>\n </semantics></math> cases of that. Catlin's classical conjecture, where <span></span><math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>=</mo>\n <mi>n</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$m=n=1$</annotation>\n </semantics></math>, follows from the classical Duffin–Schaeffer conjecture. The remaining cases of the generalized version, where <span></span><math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$m=1$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>></mo>\n <mn>1</mn>\n </mrow>\n <annotation>$n&gt;1$</annotation>\n </semantics></math>, follow from our main result. Finally, through the mass transference principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich et al.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Duffin–Schaeffer conjecture for systems of linear forms\",\"authors\":\"Felipe A. Ramírez\",\"doi\":\"10.1112/jlms.12909\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We extend the Duffin–Schaeffer conjecture to the setting of systems of <span></span><math>\\n <semantics>\\n <mi>m</mi>\\n <annotation>$m$</annotation>\\n </semantics></math> linear forms in <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math> variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math>-by-<span></span><math>\\n <semantics>\\n <mi>m</mi>\\n <annotation>$m$</annotation>\\n </semantics></math> systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n <mo>=</mo>\\n <mi>n</mi>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$m=n=1$</annotation>\\n </semantics></math>, this is the classical 1941 Duffin–Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher dimensional version, where <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n <mo>></mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$m&gt;1$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$n=1$</annotation>\\n </semantics></math>, in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010, Beresnevich and Velani proved the <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n <mo>></mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$m&gt;1$</annotation>\\n </semantics></math> cases of that. Catlin's classical conjecture, where <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n <mo>=</mo>\\n <mi>n</mi>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$m=n=1$</annotation>\\n </semantics></math>, follows from the classical Duffin–Schaeffer conjecture. The remaining cases of the generalized version, where <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$m=1$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>></mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$n&gt;1$</annotation>\\n </semantics></math>, follow from our main result. Finally, through the mass transference principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich et al.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-04-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12909\",\"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":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12909","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
我们将达芬-谢弗猜想扩展到 n 个 n$ 变量中 m 个 m$ 线性形式的系统。也就是说,我们建立了一个标准,以确定在给定的逼近率下,是否几乎所有或几乎没有 n 个 $n$ -by- m 个 $m$ 线性形式系统可以用满足自然共性条件的整数向量以该比率逼近。当 m = n = 1 $m=n=1$ 时,这就是经典的 1941 Duffin-Schaeffer 猜想,该猜想由 Koukoulopoulos 和 Maynard 于 2020 年证明。波林顿和沃恩在 1990 年证明了高维版本,即 m > 1 $m>1$和 n = 1 $n=1$。我们在此证明的一般声明是由 Beresnevich、Bernik、Dodson 和 Velani 于 2009 年猜想出来的。对于无共边性要求的近似值,他们还猜想出了卡特林猜想的广义版本,2010 年,贝雷斯内维奇和维拉尼证明了其中的 m > 1 $m>1$ 情况。卡特林的经典猜想,即 m = n = 1 $m=n=1$ ,来自经典的达芬-谢弗猜想。广义版本的其余情况,即 m = 1 $m=1$ 和 n > 1 $n>1$ ,则来自我们的主要结果。最后,通过质量转移原理,我们的主要结果暗示了它们的豪斯多夫量度类似物,这些类似物也是由 Beresnevich 等人猜想的。
The Duffin–Schaeffer conjecture for systems of linear forms
We extend the Duffin–Schaeffer conjecture to the setting of systems of linear forms in variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no -by- systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When , this is the classical 1941 Duffin–Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher dimensional version, where and , in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010, Beresnevich and Velani proved the cases of that. Catlin's classical conjecture, where , follows from the classical Duffin–Schaeffer conjecture. The remaining cases of the generalized version, where and , follow from our main result. Finally, through the mass transference principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich et al.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.