Pinned distance problem, slicing measures, and local smoothing estimates

Abstract:

We improve on 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. $y\in F$,

${\dim _{{\mathcal H}}}(\Delta ^y(E))\geq \beta$ if ${\dim _{{\mathcal H}}}(E)+\frac {d-1}{d+1}{\dim _{{\mathcal H}}}(F)>d-1+\beta$,

$\Delta ^y(E)$ has positive Lebesgue measure if ${\dim _{{\mathcal H}}}(E)+\frac {d-1}{d+1}{\dim _{{\mathcal H}}}(F)>d$,

$\Delta ^y(E)$ has nonempty interior if ${\dim _{{\mathcal H}}}(E)+\frac {d-1}{d+1}{\dim _{{\mathcal H}}}(F)>d+1$.

We also show that in the case in which ${\dim _{{\mathcal H}}}(E)+\frac {d-1}{d+1}{\dim _{{\mathcal H}}}(F)>d$, for $\mu _F$-a.e. $y\in F$, \[ \left \{t\in {\mathbb {R}} : {\dim _{{\mathcal H}}}(\{x\in E:|x-y|=t\}) \geq {\dim _{{\mathcal H}}}(E)+\frac {d+1}{d-1}{\dim _{{\mathcal H}}}(F)-d \right \} \] 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 plays a crucial role. In turn, we obtain results on sharpness of local smoothing estimates by constructing geometric counterexamples.