A Cauchy-Davenport theorem for linear maps (Q1715060)

Let \(\mathbb{F}_p\) denote the \(p\) element field for a prime \(p\). The Cauchy-Davenport theorem asserts that for \(A,B\subseteq \mathbb{F}_p\), \(\vert A+B\vert \ge \min\{p, \vert A\vert +\vert B\vert -1\}\). One might think about the set \(A+B\) as the image of the set \(A\times B\) under the map \(+: \mathbb{F}_p\times \mathbb{F}_p\rightarrow \mathbb{F}_p\). This view led the authors of the paper under review to ask for a generalization for an arbitrary linear map: let \(L: \mathbb{F}_p^n \rightarrow \mathbb{F}_p^m\) be a linear map, and \(A_1, A_2, \ldots, A_n \subseteq \mathbb{F}_p\), establish a good lower bound for \(\vert L(A_1\times A_2\times \cdots \times A_n)\vert\) in terms of \(\vert A_1\vert, \vert A_2\vert, \ldots, \vert A_n\vert\). They do find such a good lower bound, which is tight if \(m=n-1\) and \(p\) is sufficiently large. There have been related investigations for the case \(m=1\) by \textit{Z.-W. Sun} [Acta Arith. 99, No. 1, 41--60 (2001, Zbl 0974.11009)] and \textit{Z.-W. Sun} and \textit{L.-L. Zhao} [J. Comb. Theory, Ser. A 119, No. 2, 364--381 (2012, Zbl 1273.11020)], but results for \(m\geq 2\) are unprecedented. The actual bound, however, is too complex to state in a short review. The proof uses the Combinatorial Nullstellensatz and the polynomial method.