{"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}
引用次数: 0
Abstract
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.