Endpoint bounds for the non-isotropic Falconer distance problem associated with lattice-like sets

Abstract

Let $S \subset {\mathbb R}^d$ be contained in the unit ball. Let $\Delta(S)=\{||a-b||:a,b \in S\}$, the Euclidean distance set of $S$. Falconer conjectured that the $\Delta(S)$ has positive Lebesque measure if the Hausdorff dimension of $S$ is greater than $\frac{d}{2}$. He also produced an example, based on the integer lattice, showing that the exponent $\frac{d}{2}$ cannot be improved. In this paper we prove the Falconer distance conjecture for this class of sets based on the integer lattice. In dimensions four and higher we attain the endpoint by proving that the Lebesgue measure of the resulting distance set is still positive if the Hausdorff dimension of $S$ equals $\frac{d}{2}$. In three dimensions we are off by a logarithm. More generally, we consider $K$-distance sets $\Delta_K(S)=\{{|a-b|}_K: a,b \in S\}$, where ${|\cdot|}_K$ is the distance induced by a norm defined by a smooth symmetric convex body $K$ whose boundary has everywhere non-vanishing Gaussian curvature. We prove that our endpoint result still holds in this setting, providing a further illustration of the role of curvature in this class of problems.