On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$

TL;DR

This paper proves a dimension version of Falconer's distance set conjecture for sets with equal Hausdorff and packing dimensions in all dimensions, providing explicit estimates for the dimensions of distance sets in low and high dimensions.

Contribution

It establishes the dimension version of Falconer's conjecture for sets with equal Hausdorff and packing dimensions, including explicit estimates in low dimensions.

Findings

Hausdorff dimension of distance sets in the plane is at least 0.618 for sets of dimension 1

Explicit estimates for the Minkowski dimension of distance sets in higher dimensions

New estimates for dimensions of radial projections

Abstract

We establish the dimension version of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In dimensions $d=2$ or $3$, we obtain the first explicit estimates for the dimensions of distance sets of general Borel sets of dimension $d/2$; for example, we show that the set of distances spanned by a planar Borel set of Hausdorff dimension $1$ has Hausdorff dimension at least $(\sqrt{5}-1)/2\approx 0.618$. In higher dimensions we obtain explicit estimates for the lower Minkowski dimension of the distance sets of sets of dimension $d/2$. These results rely on new estimates for the dimensions of radial projections that may have independent interest.