Pinned distances of planar sets with low dimension

• Published in: arXiv.org
• Main Authors: Fiedler, Jacob B, Stull, D M
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 01.08.2024
• Subjects: Borel sets; Lower bounds
• ISSN: 2331-8422

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*}