k-plane Furstenberg sets over finite fields

Place: CUNY Graduate Center, Rm 5383

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. This 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 $k$-plane $V$ there is a $k$-plane that intersects $S$ in at least $m$ points. Using a substantial amount of algebraic geometry, Ellenberg and Erman showed that $|S| > c m^{n/k}$, for a constant $c$ depending on $n$ and $k$. In recent joint work with Manik Dhar and Zeev Dvir, we improve their bound, using much simpler proofs. For example, if $m>2^{n+7}q$, then $|S|=(1-o(1))mq^{n-k}$, where the function $o(1)$ depends on $q$. I will discuss this work, its limitations, and why the proofs of Ellenberg and Erman are still worth learning.

This is joint work with Manik Dhar and Zeev Dvir.