Dimension of Pinned Distance Sets for Semi-Regular Sets

Abstract

We prove that if $E\subseteq {\mathbb{R}}^2$ is analytic and $1<d<{\dim }_H(E)$, there are ``many'' points $x\in E$ such that the Hausdorff dimension of the pinned distance set ${\Delta }_{x}E$ is at least $d\left(1-\frac{\left(D-1\right)\left(D-d\right)}{2D^2+\left(2-4d\right)D+d^2+d-2}\right)$, where $D={\dim }_P(E)$. In particular, we prove that ${\dim }_H({\Delta }_{x}E)\ge \frac{d(d-4)}{d-5}$ for these $x$, which gives the best known lower bound for this problem when $d\in (1,5-\sqrt{15})$. We also prove that there exists some $x\in E$ such that the packing dimension of ${\Delta }_{x}E$ is at least $\frac{12-\sqrt{2}}{8\sqrt{2}}$. Moreover, whenever the packing dimension of $E$ is sufficiently close to the Hausdorff dimension of $E$, we show the pinned distance set ${\Delta }_{x}E$ has full Hausdorff dimension for many points $x\in E$; in particular the condition is that $D<\frac{(3+\sqrt{5})d-1-\sqrt{5}}{2}$. We also consider the pinned distance problem between two sets $X,Y\subseteq {\mathbb{R}}^2$, both of Hausdorff dimension greater than 1. We show that if either $X$ or $Y$ has equal Hausdorff and packing dimensions, the pinned distance ${\Delta }_{x}Y$ has full Hausdorff dimension for many points $x\in X$.