Small Nikodym sets over finite fields

A Nikodym set in $\mathbb{F}_q^d$ is a set that, from every point, contains almost a whole line through that point. Such sets must be nearly everything, but how much can be left out? For prime fields the answer is unclear even in the plane, and any construction can be checked by brute force.

Problem

Let $q$ be a prime power and $d \ge 2$. A set $N \subseteq \mathbb{F}_q^d$ is a Nikodym set if every point $x \in \mathbb{F}_q^d$ lies on some line $\ell$ with $\ell \setminus \{x\} \subseteq N$. Let $C^N(d, q)$ be the least size of a Nikodym set in $\mathbb{F}_q^d$.

• Constructions: make $q^d - C^N(d,q)$ as large as possible. In particular, for primes $p$, is $C^N(2, p) \le p^2 - c\,p^{3/2}$ for some constant $c > 0$, as it is for fields of square order?
• Lower bounds: Lund, Saraf and Wolf conjectured that $C^N(d, q) = q^d - o(q^d)$, so that Nikodym sets fill almost all of the space.

Background

Nikodym sets are the finite-field analogue of a classical object in Euclidean geometry, and are closely tied to Kakeya sets, which contain a line in every direction: a projective transformation turns a Nikodym set into essentially a Kakeya set. Kakeya sets can be as small as about $q^d/2^{d-1}$ (Bukh and Chao, 2021), while Nikodym sets are conjectured to be much larger; in the plane they are connected to blocking sets and unitals in finite projective planes. The problem is problem 1 of the AlphaEvolve repository; computer searches there led to Tao's new constructions in dimension 3 and higher.

What is known

• Plane, $q$ a perfect square: $q^2 - q^{3/2} - 1 \le C^N(2, q) \le q^2 - q^{3/2} + O(q \log q)$. The lower bound comes from the theory of blocking sets (Szőnyi et al., 2005, whose bound $q^2 - q^{3/2} - 1 + \tfrac14 s(1-s)q$ holds for all $q$, with $s$ the fractional part of $\sqrt{q}$), the upper bound from unital-based constructions (Blokhuis et al., 2015). For $q$ not a square, Georgiev, Gómez-Serrano, Tao and Wagner note that there is no obvious way to adapt these constructions.
• Dimension $d \ge 3$, $q$ odd: $C^N(d, q) \le q^d - \left(\tfrac{d-2}{\log 2} + 1 + o(1)\right) q^{d-1} \log q$ (Tao, 2026), improving the random construction's $q^d - (d - 1 + o(1))\, q^{d-1} \log q$.
• 2026 constructions: Hunter, Pohoata, Verstraete and Zhang (January 2026 preprint) improve Tao's bounds by polynomial factors, using a link with square-difference-free sets: $C^N(2, p) \le p^2 - p^{1+c}$ for primes $p$ and an absolute constant $c > 0$, the first power saving over prime fields; $C^N(3, q) \le q^3 - \Omega(q^{2.1167})$ for odd $q$; and $C^N(d, q) \le q^d - q^{d - o_d(1)}$ for prime $q$ as $d$ grows.
• Lower bounds in general: only $C^N(d, q) \ge q^d / 2^{d-1} + O(q^{d-1})$, from Bukh and Chao's sharp bounds for Kakeya sets (2021). When the characteristic stays bounded, the much stronger $q^d - O(q^{(1-\varepsilon) d})$ is known (Guo, Kopparty and Sudan, 2013).

What would count as progress

• An explicit Nikodym set in $\mathbb{F}_p^2$, $p$ prime, of size below $p^2 - c\,p^{3/2}$ for a family of primes, or for one large prime as evidence. A candidate is checked by counting, for each of the $p^2 + p$ lines, how many of its points lie outside the set; a point is covered if one of its lines misses no other point. This takes about $p^3$ operations, seconds for $p$ in the thousands.
• Larger power savings: an exponent above $1 + c$ for prime fields in the plane, where square orders suggest $3/2$, or above $2.1167$ in dimension 3.
• A proof of the Lund–Saraf–Wolf conjecture in a new range, for example $d = 3$ with unbounded characteristic.

References

1. B. Lund, S. Saraf and C. Wolf, "Finite field Kakeya and Nikodym sets in three dimensions", SIAM Journal on Discrete Mathematics 32 (2018), 2836–2849. doi:10.1137/17m1146099
2. T. Tao, "New Nikodym set constructions over finite fields", Pacific Journal of Mathematics 345 (2026), 169–186. doi:10.2140/pjm.2026.345.169, arXiv:2511.07721
3. B. Georgiev, J. Gómez-Serrano, T. Tao and A. Z. Wagner, "Mathematical exploration and discovery at scale" (2025), Problem 6.1. arXiv:2511.02864

Source

Imported from AlphaEvolve Repository of Problems, problem 1, curated by Google DeepMind. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.