Packing sets under finite groups via algebraic incidence structures

TL;DR

This paper investigates the size of the union of orbits under finite group actions on vector spaces over finite fields, establishing sharp bounds and improvements under geometric conditions, with applications to incidence geometry and sum-product problems.

Contribution

It introduces new bounds for orbit unions under group actions, extending incidence and sum-product techniques to noncommutative groups like SL_2 and the Heisenberg group.

Findings

Proved sharp bounds for orbit unions under SL_2(F_p) actions.

Established power-saving bounds under geometric non-concentration conditions.

Extended incidence and sum-product methods to noncommutative group actions.

Abstract

Let $G$ be a finite group acting on a vector space $V = \mathbb{F}_p^n$ over a prime field. Given finite sets $S \subset G$ and $E \subset V$, we study the restricted orbit union $S(E) = \bigcup_{g\in S} g(E)$ and establish quantitative lower bounds for $|S(E)|$ in terms of $|S|$, $|E|$, and natural structural conditions. This finite field packing problem has connections to distance geometry, configuration counting, and expanding graphs. For $G = SL_2(\mathbb{F}_p)$ acting on $\mathbb{F}_p^2$, we prove that $$|S(E)| \gg \min\left\lbrace p^2, \frac{|S||E|}{p^2}\right\rbrace,$$ which is sharp. Under geometric non-concentration conditions on $E$ and subgroup-avoidance hypotheses on $S$, we obtain a power-saving improvement of the form $$|S(E)|\gg \min \left\lbrace p^2, ~\max\left\lbrace\frac{|S||E|}{pk}, ~\frac{|S|^{\frac{1}{2}}|E|}{p^{\frac{1-\epsilon}{2}}k^{\frac{1}{2}}}\right\rbrace \right\rbrace,$$ where $k$ bounds the radial multiplicity of $E$. For small sets $|E| \leq p$, we establish optimal bounds using weighted incidence theory. Analogous results are proved for the first Heisenberg group $\mathbb{H}_1(\mathbb{F}_p)$ acting on $\mathbb{F}_p^3$. Our approach reformulates the problem as an incidence question in a bipartite action graph. The proofs combine Fourier analytic techniques, energy estimates, point-line incidence bounds, and area-energy inequalities for skew dot products. The methods extend classical sum-product type problems and incidence theory to noncommutative group actions.