  A $(k,m)$-Furstenberg set is a subset $S \subset \mathbb{F}_q^n$ with the
property that each $k$-dimensional subspace of $\mathbb{F}_q^n$ can be
translated so that it intersects $S$ in at least $m$ points. Ellenberg and
Erman 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 $k$. In
this paper, we adopt the same proof strategy as Ellenberg and Erman, but use
more elementary techniques than their scheme-theoretic method. By modifying
certain parts of the argument, we obtain an improved bound on $C_{n,k}$, and
our improved bound is nearly optimal for an algebraic generalization the main
combinatorial result. We also extend our analysis to give lower bounds for sets
that have large intersection with shifts of a specific family of higher-degree
co-dimension $n-k$ varieties, instead of just co-dimension $n-k$ subspaces.
