Ben Lund, Furstenberg sets over finite fields

An important family of incidence problems are discrete analogs of deep questions in geometric measure theory. Perhaps the most famous example of this is the finite field Kakeya conjecture, proved by Dvir in 2008. Dvir’s proof introduced the polynomial method to incidence geometry, which led to the solution to many long-standing problems in the area.
I will talk about a generalization of the Kakeya conjecture posed by Ellenberg, Oberlin, and Tao. A $(k,m)$-Furstenberg set S in ${\mathbb{F}}_q^n$ has the property that, parallel to every affine $k$-plane V, there is a k-plane W such that $|W\cap S|>m$. Using sophisticated ideas from algebraic geometry, Ellenberg and Erman showed that if S is a $(k,m)$-Furstenberg set, then $|S|>c{m}^{n/k}$, for a constant c depending on n and k. In recent joint work with Manik Dhar and Zeev Dvir, we give simpler proofs of stronger bounds. For example, if $m>2^{n+7}q$, then $|S|=(1-o(1))m{q}^{n-k}$, which is tight up to the $o(1)$ term.