Furstenberg sets in finite fields: Explaining and improving the Ellenberg-Erman proof

TL;DR

This paper revisits the proof of lower bounds for Furstenberg sets in finite fields, simplifying the approach and improving the bounds on their size, with extensions to higher-degree varieties.

Contribution

It provides a more elementary proof of existing bounds and improves the constant factors, extending results to intersections with higher-degree varieties.

Findings

Improved lower bounds for Furstenberg sets in finite fields.

Elementary proof techniques replacing scheme-theoretic methods.

Extended bounds to intersections with higher-degree varieties.

Abstract

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.