Evasive sets, covering by subspaces, and point-hyperplane incidences

TL;DR

This paper investigates the size and properties of subspace evasive sets over finite fields, providing new bounds and proofs, and explores their implications in combinatorial geometry, including hyperplane coverings and point-hyperplane incidences.

Contribution

It offers an alternative proof for the size bounds of subspace evasive sets using the random algebraic method and extends results to large dimensions, with applications in geometry.

Findings

Established sharp upper bounds for large-dimensional evasive sets.

Proved the minimum number of hyperplanes to cover a grid matches known upper bounds.

Improved lower bounds on point-hyperplane incidences under certain graph constraints.

Abstract

Given positive integers $k\leq d$ and a finite field $\mathbb{F}$, a set $S\subset\mathbb{F}^{d}$ is $(k,c)$-subspace evasive if every $k$-dimensional affine subspace contains at most $c$ elements of $S$. By a simple averaging argument, the maximum size of a $(k,c)$-subspace evasive set is at most $c |\mathbb{F}|^{d-k}$. When $k$ and $d$ are fixed, and $c$ is sufficiently large, the matching lower bound $\Omega(|\mathbb{F}|^{d-k})$ is proved by Dvir and Lovett. We provide an alternative proof of this result using the random algebraic method. We also prove sharp upper bounds on the size of $(k,c)$-evasive sets in case $d$ is large, extending results of Ben-Aroya and Shinkar. The existence of optimal evasive sets has several interesting consequences in combinatorial geometry. We show that the minimum number of $k$-dimensional linear hyperplanes needed to cover the grid $[n]^{d}\subset \mathbb{R}^{d}$ is $\Omega_{d}\big(n^{\frac{d(d-k)}{d-1}}\big)$, which matches the upper bound proved by Balko, Cibulka, and Valtr, and settles a problem proposed by Brass, Moser, and Pach. Furthermore, we improve the best known lower bound on the maximum number of incidences between points and hyperplanes in $\mathbb{R}^{d}$ assuming their incidence graph avoids the complete bipartite graph $K_{c,c}$ for some large constant $c=c(d)$.