Joint BCAM-UPV/EHU Analysis and PDE seminar: A singular variant of the Falconer distance problem

Abstract
In this talk we will discuss the following variant of the Falconer distance problem. Let $E$ be a compact subset of ${mathbb{R}}^d$, $d geq 1$, and define $$ Box(E)={|(y,z)-(x,x)|: (y,z)in Etimes E,x in E,, yneq z }subseteq mathbb{R}.$$ This is the set of distances between points of $Etimes E $ and the diagonal $mathcal{D}_{Etimes E}={(x,x)colon xin E}$ with the additional non-degeneracy condition $yneq z$.

We showed using a variety of methods that if the Hausdorff dimension of $E$ is greater than $frac{d}{2}+frac{1}{4}$, then the Lebesgue measure of $Box(E)$ is positive. This problem can be viewed as a singular variant of the classical Falconer distance problem because considering the diagonal $(x,x)$ in the definition of $Box(E)$ poses interesting complications stemming from the fact that the set ${(x,x): x in E}subseteq mathbb{R}^{2d}$ is much smaller than the sets for which the Falconer type results are typically established.

This talk is based on joint work with Alex Iosevich and Yumeng Ou.

More info at https://sites.google.com/view/apdebilbao/home