Furstenberg sets and Furstenberg schemes over finite fields

Abstract

We give a lower bound for the size of a subset of ${\mathbb{F}}_q^n$ containing a rich $k$-plane in every direction, a $k$-plane Furstenberg set. The chief novelty of our method is that we use arguments on nonreduced subschemes and flat families to derive combinatorial facts about incidences between points and $k$-planes in space.