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 $E, F\subset{\Bbb R}^d$, 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.