Furstenberg sets and Furstenberg schemes over finite fields

TL;DR

This paper establishes a lower bound on the size of Furstenberg sets in finite fields using novel algebraic geometry techniques involving non-reduced schemes and flat families to analyze incidences.

Contribution

It introduces a new method employing algebraic geometry tools to study combinatorial incidence problems over finite fields, providing bounds for Furstenberg sets.

Findings

Derived a lower bound for Furstenberg sets in finite fields.

Utilized non-reduced subschemes and flat families in combinatorial geometry.

Established a novel algebraic approach to incidence problems.

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 non-reduced subschemes and flat families to derive combinatorial facts about incidences between points and k-planes in space.