Furstenberg sets in finite fields: Explaining and improving the Ellenberg-Erman proof

A subset $S subset mathbb{F}_q^n$, where $mathbb{F}_q$ is a finite field, is called $(k,m)$-Furstenberg if it has $m$ common points with a $k$-flat in each direction. That is, any $k$-dimensional subspace of $mathbb{F}_q^n$ can be translated so that it intersects $S$ in at least $m$ points. Using sophisticated scheme-theoretic machinery, Ellenberg and Erman proved that $(k,m)$-Furstenberg sets must have size at least $C_{n,k}m^{n/k}$ with a constant $C_{n,k}$ depending only $n$ and $k$. In this work we follow the overall proof strategy of Ellenberg-Erman, replacing the scheme-theoretic language with more elementary machinery. In addition to presenting the proof in a self-contained and accessible form, we are also able to improve the constant $C_{n,k}$ by modifying certain key parts of the argument.