Given a compact subset $E$ of Euclidean space of dimension $s>\frac{d}{2}$, and a Frostman measure $\mu$ on this set, we are going to consider operators of the form $T_Kf(x)=\int K(x,y) f(y) d\mu(y),$$ where $K$ is a non-negative $L^1$ kernel or a compactly supported measure. We form a distance graph by taking the points of $E$ as vertices and connecting two vertices $x,y$ by an edge if $K(x,y)>0$. We are going to see that many of the classical distance set techniques go through in this setting. We will also see that this point of view allows us to recover configurations that would have been difficult to achieve with classical methods.

Password: 00089