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A $(k,m)$-Furstenberg set is a subset $S \subset \mathbb{F}_q^n$ with theproperty that each $k$-dimensional subspace of $\mathbb{F}_q^n$ can betranslated so that it intersects $S$ in at least $m$ points. Ellenberg andErman proved that $(k,m)$-Furstenberg sets must have size at least$C_{n,k}m^{n/k}$, where $C_{n,k}$ is a constant depending only $n$ and $k$. Inthis paper, we adopt the same proof strategy as Ellenberg and Erman, but usemore elementary techniques than their scheme-theoretic method. By modifyingcertain parts of the argument, we obtain an improved bound on $C_{n,k}$, andour improved bound is nearly optimal for an algebraic generalization the maincombinatorial result. We also extend our analysis to give lower bounds for setsthat have large intersection with shifts of a specific family of higher-degreeco-dimension $n-k$ varieties, instead of just co-dimension $n-k$ subspaces.

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