Year: 2026
Type: article
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(Δ_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(Δ_x Y) = \min\{\dim_H(Y), 1\}. \end{equation*}
DOI: 10.1090/tran/9897
Authors: Jacob Fiedler, D. Stull
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4401978558