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 $$\mathbb{F}_p $$Fp, 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 $$L:\mathbb{F}_p^n \to \mathbb{F}_p^m $$L:Fpn→Fpm, and subsets $$A_1 , \ldots A_n \subseteq \mathbb{F}_p$$A1,…An⊆Fp, and gives a lower bound on the size of L(A1 × A2 × … × An) in terms of the sizes of the sets A1, …, An.Our proof uses Alon’s Combinatorial Nullstellensatz and a variation of the polynomial method.