📝 SummarXiv

Abstract

Let $E$ be a set in $\mathbb{F}_p^n$ and $S$ be a set of maps from $\mathbb{F}_p^n$ to $\mathbb{F}_p^n$. We define \[ S (E) := \bigcup_{f\in S} f(E) = \left\lbrace f(x) \colon x\in E, f\in S \right\rbrace.\] In this paper, we establish sharp lower bounds on the size of $S(E)$ when $S$ consists of matrices from either the special linear group $SL_2(\mathbb{F}_p)$ or the first Heisenberg group $\mathbb{H}_1(\mathbb{F}_p)$. Our proofs are based on novel results on algebraic incidence-type structures associated with these groups. We also discuss higher-dimensional generalizations.