Large point-line matchings and small Nikodym sets

TL;DR

This paper constructs large point-line matchings in finite field geometries using a novel connection to arithmetic combinatorics, leading to improved bounds for Nikodym sets, blocking sets, and minimal distance problems.

Contribution

It introduces new large matchings in finite field incidence graphs and applies these to improve bounds on Nikodym sets, blocking sets, and minimal distance configurations.

Findings

Constructed matchings of size q^{1.233} in _q^2

Improved Nikodym set sizes to q^d - q^{d - o_d(1)}

Solved a longstanding problem by constructing a minimal blocking set in _qP^2

Abstract

For any integer $d \geq 2$ and prime power $q$, we construct unexpectedly large induced matchings in the point-line incidence graph of $\mathbb{F}_{q}^{d}$ by leveraging a new connection with the Furstenberg-S\'ark\"ozy problem from arithmetic combinatorics. In particular, we significantly improve the previously well-known baselines when $q$ is prime, showing that $\mathbb{F}_{q}^{2}$ contains matchings of size $q^{1.233}$ and $\mathbb{F}_{q}^{d}$ contains matchings of size $q^{d-o_{d}(1)}$. These results and their proofs have several applications. First, we also obtain new constructions for finite field Nikodym sets in dimension $d \geq 2$, improving recent results of Tao by polynomial factors. For example, when $q$ is prime, we show the existence of Nikodym sets in $\mathbb{F}_q^d$ of size $q^d - q^{d - o_d(1)}$. Second, we construct a new minimal blocking set in $\mathrm{PG}(2,q)$, solving a longstanding problem in finite geometry. Third, we obtain new constructions for the minimal distance problem (in $\mathbb{R}^{2}$ and also in higher dimensions), improving a recent result of Logunov-Zakharov. We also obtain analogous results for general finite fields with large characteristics. In particular, in one of our constructions we introduce a new special set of points inside the norm hypersurface in $\mathbb{F}_{q}^{d}$, which directly generalizes the classical Hermitian unital and which may be of independent interest for applications.