A Cauchy-Davenport theorem for linear maps

We prove a version of the Cauchy-Davenport theorem for general linear maps. For subsets $A,B$ of the finite field $\mathbb{F}_p$, the classical Cauchy-Davenport theorem gives a lower bound for the size of the sumset $A+B$ in terms of the sizes of the sets $A$ and $B$. Our theorem considers a general linear map $L: \mathbb{F}_p^n \to \mathbb{F}_p^m$, and subsets $A_1, \ldots, A_n \subseteq \mathbb{F}_p$, and gives a lower bound on the size of $L(A_1 \times A_2 \times \ldots \times A_n)$ in terms of the sizes of the sets $A_1, \ldots, A_n$. Our proof uses Alon's Combinatorial Nullstellensatz and a variation of the polynomial method.