{"title":"Extensions of Schreiber’s theorem on discrete approximate subgroups in $\\protect \\mathbb{R}^d$","authors":"A. Fish","doi":"10.5802/JEP.90","DOIUrl":null,"url":null,"abstract":"In this paper we give an alternative proof of Schreiber's theorem which says that an infinite discrete approximate subgroup in $\\mathbb{R}^d$ is relatively dense around a subspace. We also deduce from Schreiber's theorem two new results. The first one says that any infinite discrete approximate subgroup in $\\mathbb{R}^d$ is a restriction of a Meyer set to a thickening of a linear subspace in $\\mathbb{R}^d$, and the second one provides an extension of Schreiber's theorem to the case of the Heisenberg group.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"10 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de l’École polytechnique — Mathématiques","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/JEP.90","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 8
Abstract
In this paper we give an alternative proof of Schreiber's theorem which says that an infinite discrete approximate subgroup in $\mathbb{R}^d$ is relatively dense around a subspace. We also deduce from Schreiber's theorem two new results. The first one says that any infinite discrete approximate subgroup in $\mathbb{R}^d$ is a restriction of a Meyer set to a thickening of a linear subspace in $\mathbb{R}^d$, and the second one provides an extension of Schreiber's theorem to the case of the Heisenberg group.