On Densities in the Erdős-Falconer Distance Problems over the Integers
Motivated by the classical Erdős and Falconer distance problems and Magyar's work on distance sets in $\mathbb Z^d$, we study discrete analogues of the Erdős-Falconer unpinned and pinned distance problems over the integers. We ask how the discrete Hausdorff dimension of a set $A\subseteq\mathbb Z^d$ influences the size of its squared-distance sets, measured by generalized upper natural density and upper discrete mass dimension. We propose conjectures for both problems and prove the unpinned conjecture in dimensions $d=1,2,3$ and the pinned conjecture in dimensions $d=1,2$. For the unpinned problem in higher dimensions, we obtain explicit lower bounds for the dimensions of distance sets using elementary integer sphere counting, results on the Erdős distinct distances problem, and $\ell^p$-improving inequalities for discrete spherical averages. For the pinned problem, we obtain lower bounds for the dimensions of distance sets for some or every pin. We also establish more general results involving upper discrete mass dimension, upper discrete Banach dimension, and affine dimension, including bounds on the affine dimension of the exceptional set of pins. We construct examples showing that both conjectures are sharp in low dimensions. Further examples demonstrate how the distinctions among discrete Hausdorff dimension, upper discrete mass dimension, and upper discrete Banach dimension lead to different phenomena in the unpinned and pinned distance problems, including limitations on the nontrivial upper bounds these dimensions can provide for the dimensions of distance sets.