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

Date: Thu, Jun 29 2023

Hour: 17:00

Location: Maryam Mirzakhani Seminar Room at BCAM

Speakers: Tainara Borges (she/her) (Brown University)

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