Pinned distance problem, slicing measures and local smoothing estimates

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 $E,F\subset {\mathbb{R}}^d$, there exists a probability measure ${\mu }_F$ on $F$ such that for ${\mu }_F$-a.e. $y\in F$, (1) ${\dim }_{\mathcal{H}}({\Delta }^y(E))\ge \beta$ 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. $y\in F$, $$\left\{t\in \mathbb{R}:{\dim }_{\mathcal{H}}(\{x\in E:|x-y|=t\})\ge {\dim }_{\mathcal{H}}(E)+\frac{d+1}{d-1}{\dim }_{\mathcal{H}}(F)-d\right\}$$ 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.

Cited by in corpus (2)

• New bounds on the dimensions of planar distance sets
• On Falconer's distance set problem in the plane