A singular variant of the Falconer distance problem

In this paper we study the following variant of the Falconer distance problem. Let $E$ be a compact subset of ${\mathbb{R}}^d$, $d \ge 1$, and define $$\Box(E)=\left\{\sqrt{{|x-y|}^2+{|x-z|}^2}: x,y,z \in E,\, y\neq z \right\}.$$ We shall prove 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. We also prove a finite field variant of the Euclidean results for $\Box(E)$ and indicate both the similarities and the differences between the two settings.