Bounds on sizes of general caps in $AG(n,q)$ via the Croot-Lev-Pach polynomial method

Abstract

In 2016, Ellenberg and Gijswijt employed a method of Croot, Lev, and Pach to show that a maximal cap in $AG(n,q)$ has size $O(q^{cn})$ for some $c<1$. In this paper, we show more generally that if $S$ is a subset of $AG(n,q)$ containing no $m$ points on any $(m-2)$- flat, then $|S|<q^{c_{m}n}$ for some $c_m<1$, as long as $q$ is odd or $m$ is even.

15 pages