Pinned distance problem, slicing measures and local smoothing estimates

TL;DR

This paper advances the understanding of pinned distance sets in Euclidean spaces, establishing new dimension and measure results using Fourier integral operator smoothing estimates and geometric counterexamples.

Contribution

It improves previous results on pinned distances by linking Hausdorff dimensions with measure and interior properties, employing local smoothing estimates of Fourier integral operators.

Findings

Pinned distance sets have positive Lebesgue measure under certain dimension conditions.

The dimension of sliced subsets of E can be characterized using local smoothing estimates.

Constructed geometric counterexamples demonstrate sharpness of the results.

Abstract

We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$\Delta^y(E) = \{|x-y|:x\in E\},$$ we prove that for any $E, F\subset{\Bbb R}^d$, there exists a probability measure $\mu_F$ on $F$ such that for $\mu_F$-a.e. $y\in F$, (1) $\dim_{{\mathcal H}}(\Delta^y(E))\geq\beta$ if $\dim_{{\mathcal H}}(E) + \frac{d-1}{d+1}\dim_{{\mathcal H}}(F) > d - 1 + \beta$; (2) $\Delta^y(E)$ has positive Lebesgue measure if $\dim_{{\mathcal H}}(E)+\frac{d-1}{d+1}\dim_{{\mathcal H}}(F) > d$; (3) $\Delta^y(E)$ has non-empty interior if $\dim_{{\mathcal H}}(E)+\frac{d-1}{d+1}\dim_{{\mathcal H}}(F) > d+1$. We also show that in the case when $\dim_{{\mathcal H}}(E)+\frac{d-1}{d+1}\dim_{{\mathcal H}}(F)>d$, for $\mu_F$-a.e. $y\in F$, $$ \left\{t\in{\Bbb R} : \dim_{{\mathcal H}}(\{x\in E:|x-y|=t\}) \geq \dim_{{\mathcal H}}(E)+\frac{d+1}{d-1}\dim_{{\mathcal H}}(F)-d \right\} $$ has positive Lebesgue measure. This describes dimensions of slicing subsets of $E$, sliced by spheres centered at $y$. In our proof, local smoothing estimates of Fourier integral operators (FIO) plays a crucial role. In turn, we obtain results on sharpness of local smoothing estimates by constructing geometric counterexamples.