# On the Hausdorff dimension of pinned distance sets

## TL;DR

This paper proves that for certain planar sets with equal Hausdorff and packing dimension greater than 1, the set of distances from most points in the set has full Hausdorff dimension, confirming a strong version of Falconer's conjecture.

## Contribution

It establishes a new result confirming a strong variant of Falconer's distance set conjecture for sets with equal Hausdorff and packing dimension greater than 1.

## Key findings

- Pinned distance sets have full Hausdorff dimension for most points in the set.
- Verifies a strong variant of Falconer's distance set conjecture.
- Applies to sets with equal Hausdorff and packing dimension s>1.

## Abstract

We prove that if $A$ is a Borel set in the plane of equal Hausdorff and packing dimension $s>1$, then the set of pinned distances $\{ |x-y|:y\in A\}$ has full Hausdorff dimension for all $x$ outside of a set of Hausdorff dimension $1$ (in particular, for many $x\in A$). This verifies a strong variant of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension, outside the endpoint $s=1$.