{"title":"The Furstenberg set and its random version","authors":"A. Fan, Herv'e Queff'elec, M. Queff'elec","doi":"10.4171/lem/1040","DOIUrl":null,"url":null,"abstract":"We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers $S=\\{2^{m}3^{n}\\}$ and compare them to those of its random analogue $T$. In this half-expository work, we show for example that $S$ is \"Khinchin distributed\", is far from being Hartman-distributed while $T$ is, and that $S$ is a $\\Lambda(p)$ set for all $2<p<\\infty$ and that $T$ is a $p$-Rider set for all $p$ such that $4/3<p<2$. Measure-theoretic and probabilistic techniques, notably martingales, play an important role in this work.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"10 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"L’Enseignement Mathématique","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4171/lem/1040","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers $S=\{2^{m}3^{n}\}$ and compare them to those of its random analogue $T$. In this half-expository work, we show for example that $S$ is "Khinchin distributed", is far from being Hartman-distributed while $T$ is, and that $S$ is a $\Lambda(p)$ set for all $2
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