Sets preserved by a large subgroup of the special linear group

Abstract

Let $E$ be a subset of the affine plane over a finite field ${\mathbb{F}}_q$. We bound the size of the subgroup of $S{L}_2({\mathbb{F}}_q)$ that preserves $E$. As a consequence, we show that if $E$ has size $\ll q^{\alpha }$ and is preserved by $\gg q^{\beta }$ elements of $S{L}_2({\mathbb{F}}_q)$ with $\beta \ge 3\alpha /2$, then $E$ is contained in a line. This result is sharp in general, and will be proved by using combinatorial arguments and applying a point-line incidence bound in ${\mathbb{F}}_q^3$ due to Mockenhaupt and Tao (2004).

arXiv commentV2: Proposition 1.6 over prime fields is added, typos corrected, 10 pages