{"title":"Product-free sets in the free group","authors":"Miquel Ortega, Juanjo Rué, Oriol Serra","doi":"10.1112/mtk.12255","DOIUrl":null,"url":null,"abstract":"<p>We prove that product-free subsets of the free group over a finite alphabet have maximum upper density <span></span><math></math> with respect to the natural measure that assigns total weight one to each set of irreducible words of a given length. This confirms a conjecture of Leader, Letzter, Narayanan, and Walters. In more general terms, we actually prove that strongly <span></span><math></math>-product-free sets have maximum upper density <span></span><math></math> in terms of this measure. The bounds are tight.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12255","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that product-free subsets of the free group over a finite alphabet have maximum upper density with respect to the natural measure that assigns total weight one to each set of irreducible words of a given length. This confirms a conjecture of Leader, Letzter, Narayanan, and Walters. In more general terms, we actually prove that strongly -product-free sets have maximum upper density in terms of this measure. The bounds are tight.
期刊介绍：
Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.
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