Improved bounds on sizes of generalized caps in $AG(n,q)$

An $m$-general set in $AG(n,q)$ is a set of points such that any subset of size $m$ is in general position. A $3$-general set is often called a capset. In this paper, we study the maximum size of an $m$-general set in $AG(n,q)$, significantly improving previous results. When $m=4$ and $q=2$ we give a precise estimate, solving a problem raised by Bennett.