Large point-line matchings and small Nikodym sets

TL;DR

The paper develops new lower bounds for induced matchings in point–line incidence graphs over finite fields, notably proving $\mathrm{IM}(2,q) \gg q^{1.2334}$ for all primes $q$ and $\mathrm{IM}(d,q) \gg_d q^{d-\varepsilon_d}$ with $\varepsilon_d \ll (\log d)^{-1}$, by building large matchings via lifting constructions from Paley graphs, Ruzsa sets, and norm-hypersurfaces. It then translates these matchings into sharp finite-field Nikodym-set results, yielding Nikodym sets in $\mathbb{F}_q^d$ of size $q^d - q^{d-o_d(1)}$ for prime $q$, and, in parallel, polynomially larger minimal blocking sets in $\mathrm{PG}(2,q)$. The work further connects to the minimal-distance problem through a new mechanism that transfers high-dimensional point–line configurations to Euclidean $d_\infty$-distance statements, proving, among other things, that $\mathrm{PL}_2(0.7666)$ is false and that $\mathrm{PL}_d(\gamma)$ fails for appropriate $d$ depending on $\gamma$. In addition to expanding the range of known constructions, the results reveal deep links between incidence geometry, additive combinatorics (notably Furstenberg–Sárközy phenomena), and finite-geometric objects such as norm hypersurfaces and Hermitian unitals, with potential implications for other extremal problems.

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-Sárközy 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.

Figures (1)

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Theorems & Definitions (78)

• Definition 1.1: Induced matchings
• Theorem 1.2
• Definition 1.3: Nikodym set
• Definition 1.4: Weak Nikodym set
• Proposition 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10