Pinned distances of planar sets with low dimension

TL;DR

This paper improves bounds on the Hausdorff dimension of pinned distance sets for planar sets with low dimension and explores the existence of universal sets with maximal pinned distance dimensions.

Contribution

It provides new bounds for the Hausdorff dimension of pinned distance sets of low-dimensional planar sets and establishes the existence of universal sets with maximal pinned distance dimensions.

Findings

Improved bounds on Hausdorff dimension of pinned distance sets.

Existence of small universal sets for pinned distances.

Maximal pinned distance dimension for certain regular sets.

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 Ahlfors-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*}