{"title":"Absolute continuity of Bernoulli convolutions for algebraic parameters","authors":"P. P. Varj'u","doi":"10.1090/jams/916","DOIUrl":null,"url":null,"abstract":"<p>We prove that Bernoulli convolutions <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu Subscript lamda\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>μ<!-- μ --></mml:mi>\n <mml:mi>λ<!-- λ --></mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mu _\\lambda</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are absolutely continuous provided the parameter <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\">\n <mml:semantics>\n <mml:mi>λ<!-- λ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\lambda</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is an algebraic number sufficiently close to <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> depending on the Mahler measure of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\">\n <mml:semantics>\n <mml:mi>λ<!-- λ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\lambda</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.5000,"publicationDate":"2016-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/916","citationCount":"42","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jams/916","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 42
Abstract
We prove that Bernoulli convolutions μλ\mu _\lambda are absolutely continuous provided the parameter λ\lambda is an algebraic number sufficiently close to 11 depending on the Mahler measure of λ\lambda.
期刊介绍:
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