{"title":"Pinned distances of planar sets with low dimension","authors":"Jacob B. Fiedler, D. M. Stull","doi":"arxiv-2408.00889","DOIUrl":null,"url":null,"abstract":"In this paper, we give improved bounds on the Hausdorff dimension of pinned\ndistance sets of planar sets with dimension strictly less than one. As the\nplanar set becomes more regular (i.e., the Hausdorff and packing dimension\nbecome closer), our lower bound on the Hausdorff dimension of the pinned\ndistance set improves. Additionally, we prove the existence of small universal\nsets for pinned distances. In particular, we show that, if a Borel set\n$X\\subseteq\\mathbb{R}^2$ is weakly regular ($\\dim_H(X) = \\dim_P(X)$), and\n$\\dim_H(X) > 1$, then \\begin{equation*} \\sup\\limits_{x\\in X}\\dim_H(\\Delta_x Y) = \\min\\{\\dim_H(Y), 1\\} \\end{equation*} for every Borel set $Y\\subseteq\\mathbb{R}^2$. Furthermore, if $X$ is also\ncompact and Alfors-David regular, then for every Borel set\n$Y\\subseteq\\mathbb{R}^2$, there exists some $x\\in X$ such that \\begin{equation*} \\dim_H(\\Delta_x Y) = \\min\\{\\dim_H(Y), 1\\}. \\end{equation*}","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.00889","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we give improved bounds on the Hausdorff dimension of pinned
distance sets of planar sets with dimension strictly less than one. As the
planar set becomes more regular (i.e., the Hausdorff and packing dimension
become closer), our lower bound on the Hausdorff dimension of the pinned
distance set improves. Additionally, we prove the existence of small universal
sets for pinned distances. In particular, we show that, if a Borel set
$X\subseteq\mathbb{R}^2$ is weakly regular ($\dim_H(X) = \dim_P(X)$), and
$\dim_H(X) > 1$, then \begin{equation*} \sup\limits_{x\in X}\dim_H(\Delta_x Y) = \min\{\dim_H(Y), 1\} \end{equation*} for every Borel set $Y\subseteq\mathbb{R}^2$. Furthermore, if $X$ is also
compact and Alfors-David regular, then for every Borel set
$Y\subseteq\mathbb{R}^2$, there exists some $x\in X$ such that \begin{equation*} \dim_H(\Delta_x Y) = \min\{\dim_H(Y), 1\}. \end{equation*}