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 $$ \Box(E)=\{|(y,z)-(x,x)|: (y,z)\in E\times E,x \in E,\, y\neq z \}\subseteq \mathbb{R}.$$ This is the set of distances between points of $E\times E $ and the diagonal $\mathcal{D}_{E\times E}=\{(x,x)\colon x\in E\}$ with the additional non-degeneracy condition $y\neq z$. 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.