Dimension of Pinned Distance Sets for Semi-Regular Sets

TL;DR

This paper establishes new lower bounds for the Hausdorff and packing dimensions of pinned distance sets for semi-regular sets in the plane, advancing understanding of distance set problems in geometric measure theory.

Contribution

It provides the best known lower bounds for the Hausdorff dimension of pinned distance sets when the dimension is between 1 and approximately 1.686, and explores conditions for full dimension in related settings.

Findings

Lower bound for Hausdorff dimension of pinned distance sets when d in (1, 5−√15)

Existence of points with pinned distance set of large packing dimension

Conditions under which pinned distance sets have full Hausdorff dimension

Abstract

We prove that if $E\subseteq \R^2$ 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 $\dim_H(\Delta_x E) \geq \frac{d(d-4)}{d-5}$ 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 $X, Y\subseteq \R^2$, 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 $x\in X$.