{"title":"Large dilates of hypercube graphs in the plane","authors":"V. Kovač, B. Predojević","doi":"10.1007/s10476-024-00045-6","DOIUrl":null,"url":null,"abstract":"<div><p>We study a distance graph <span>\\(\\Gamma_n\\)</span> that is isomorphic to the <span>\\(1\\)</span>-skeleton of an <span>\\(n\\)</span>-dimensional unit hypercube. We show that every measurable set of positive upper Banach density in the plane contains all sufficiently large dilates of <span>\\(\\Gamma_n\\)</span>. This provides the first examples of distance graphs other than the trees for which a dimensionally sharp embedding in positive density sets is known.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":"50 3","pages":"893 - 915"},"PeriodicalIF":0.6000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis Mathematica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10476-024-00045-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study a distance graph \(\Gamma_n\) that is isomorphic to the \(1\)-skeleton of an \(n\)-dimensional unit hypercube. We show that every measurable set of positive upper Banach density in the plane contains all sufficiently large dilates of \(\Gamma_n\). This provides the first examples of distance graphs other than the trees for which a dimensionally sharp embedding in positive density sets is known.
期刊介绍：
Traditionally the emphasis of Analysis Mathematica is classical analysis, including real functions (MSC 2010: 26xx), measure and integration (28xx), functions of a complex variable (30xx), special functions (33xx), sequences, series, summability (40xx), approximations and expansions (41xx).
The scope also includes potential theory (31xx), several complex variables and analytic spaces (32xx), harmonic analysis on Euclidean spaces (42xx), abstract harmonic analysis (43xx).
The journal willingly considers papers in difference and functional equations (39xx), functional analysis (46xx), operator theory (47xx), analysis on topological groups and metric spaces, matrix analysis, discrete versions of topics in analysis, convex and geometric analysis and the interplay between geometry and analysis.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
