Inverse Theorems for Point-Sphere Incidences over Finite Fields

Abstract

We prove the first inverse theorem for point--sphere incidence bounds over finite fields in dimensions $d \ge 3$, showing that near-extremality forces algebraic rigidity. While sharp upper bounds have been known for over a decade, the structural characterization of configurations that nearly saturate these bounds has remained completely open. Specifically, if a configuration of points $P \subset \mathbb{F}_q^d$ and spheres $\mathscr{S}$ exceeds the random incidence baseline by a factor $K$ in the moderate-sphere regime, then there exists a subset $P' \subset P$ of size \[ |P'| \gtrsim K q^{(d-1)/2} \] contained in the zero set of a polynomial $F$ of degree at most $C K^C$. This yields a one-sided result: we identify necessary algebraic obstructions to extremality, without asserting sufficiency. The proof introduces a new rigidity mechanism for finite-field incidence geometry. Near-extremality manifests as persistent overlap among bisector hyperplanes. We prove that such persistent coincidence cannot occur without forcing the emergence of bounded-complexity algebraic certificates. The argument proceeds by isolating high-overlap layers via energy stratification, followed by a projective polynomial dichotomy applied to the set of normal directions. As applications, we obtain the first inverse-type results for pinned distance and dot-product problems over finite fields, resolving structural questions inaccessible to standard polynomial or Fourier-analytic methods.