Sphere intersections and incidences over finite fields
Doowon Koh, Ben Lund, Chuandong Xu, Semin Yoo
TL;DR
This work addresses incidences between points and spheres over finite fields by bounding the incidences via pairwise intersections of spheres, using a pseudo-random graph framework. A general weighted incidence bound ties the deviation $I(P,S)-p|P||S|$ to the sum of pairwise intersection sizes and sphere sizes, enabling explicit bounds in both even and odd dimensions. In even dimensions, nonzero-radius spheres yield sharp bounds of the form $|I(P,S)-q^{-1}|P||S|| \le (1+o(1))\sqrt{|P|(q^{d-1}|S| + q^{d/2-1}|S|^2)}$, while odd dimensions admit an elementary bound $|I(P,S)-q^{-1}|P||S|| \le q^{(d-1)/2}|P|^{1/2}|S|^{1/2} + \sqrt{3}q^{(d-3)/4}|P|^{1/2}|S|$ for radii from restricted sets, along with an elementary proof of the odd-dimensional Iosevich–Rudnev bound. The results hinge on explicit calculations of pairwise sphere intersections, $|s_1 \cap s_2| = q^{d-2} + N_2(Q,r_1,r_2,t)$, for general non-degenerate quadratic forms $Q$, and have implications for the Erdős–Falconer distance problem in odd and certain even dimensions. Overall, the paper provides a coherent framework linking sphere intersections to incidence bounds and delivers improvements in regimes with relatively few spheres, supported by a detailed treatment of quadratic forms and intersection counts.
Abstract
We bound the number of incidences between points and spheres in finite vector spaces by bounding the sum of the number of points in the pairwise intersections of the spheres. We obtain new incidence bounds that are interesting when the number of spheres is not too large. Our approach also leads to an elementary proof of the Iosevich-Rudnev bound on the Erdős-Falconer distance problem in odd dimensions.