Tinkering with Lattices: A New Take on the Erd\H{o}s Distance Problem

The Erd\H{o}s distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \textit{near-optimal}, as it spans around $O(N/\sqrt{\log(N)})$ distinct distances which is the lower bound for a set of $N$ points (Erd\H{o}s, 1946). The only previous non-asymptotic work relating to the Erd\H{o}s distance problem that has been done was carried out for $N \leq 13$. We take a new non-asymptotic approach to this problem, studying the distance distribution, or in other words, the plot of frequencies of each distance of the $N\times N$ integer lattice. In order to fully characterize this distribution and determine its most common and least common distances, we adapt previous number-theoretic results from Fermat and Erd\H{o}s, in order to relate the frequency of a given distance on the lattice to the sum-of-squares formula. We study the distance distributions of all its possible subsets; although this is a restricted case, we find that the structure of the integer lattice allows for the existence of subsets which can be chosen so that their distance distributions have certain properties, such as emulating the distribution of randomly distributed sets of points for certain small subsets, or that of the larger lattice itself. We define an error which compares the distance distribution of a subset with that of the full lattice. The structure of the integer lattice allows us to take subsets with certain geometric properties in order to maximize error, by exploiting the potential for sub-structure in the integer lattice. We show these geometric constructions explicitly; further, we calculate explicit upper bounds for the error for when the number of points in the subset is $4$, $5$, $9$ or $\left \lceil N^2/2\right\rceil$ and prove a lower bound for more general numbers of points.