Pinned distances of planar sets with low dimension

Jacob B. Fiedler, D. M. Stull
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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*}
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低维度平面集的钉距
在本文中,我们给出了维度严格小于 1 的平面集的钉距集的豪斯多夫维度的改进边界。随着平面集变得越来越规则(即 Hausdorff 维度和堆积维度越来越接近),我们对钉扎距离集的 Hausdorff 维度的下界也随之提高。此外,我们还证明了钉住距离小普遍集的存在。我们特别证明了,如果一个波尔集合$X(subseteq/mathbb{R}^2$是弱正则的($\dim_H(X) = \dim_P(X)$),并且$\dim_H(X) > 1$,那么}\sup\limits_{x\in X}\dim_H(\Delta_x Y) = \min\{\dim_H(Y), 1\}\end{equation*} for every Borel set $Y\subseteq\mathbb{R}^2$.此外,如果 $X$ 也是紧凑的、Alfors-David 正则的,那么对于每个 Borel 集$Y(subseteq\mathbb{R}^2$),在 X$ 中存在一些 $x\in ,使得 \begin{equation*}\dim_H(\Delta_x Y) = \min\{dim_H(Y), 1\}.\end{equation*}
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