Large point-line matchings and small Nikodym sets
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| Format: | Preprint |
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2026
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| _version_ | 1866912853568520192 |
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| author | Hunter, Zach Pohoata, Cosmin Verstraete, Jacques Zhang, Shengtong |
| author_facet | Hunter, Zach Pohoata, Cosmin Verstraete, Jacques Zhang, Shengtong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19879 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large point-line matchings and small Nikodym sets Hunter, Zach Pohoata, Cosmin Verstraete, Jacques Zhang, Shengtong Combinatorics Number Theory 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. |
| title | Large point-line matchings and small Nikodym sets |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2601.19879 |