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

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 $|S| < q^{c_mn}$ for some $c_m < 1$, as long as $q$ is odd or $m$ is even.