New Nikodym set constructions over finite fields (terrytao.wordpress.com)

The author posted to arXiv a new paper giving improved constructions of Nikodym sets over finite fields — a follow-up to experiments that used AI tools (AlphaEvolve, DeepThink, DeepThink/AlphaProof variants) to explore combinatorial-geometric constructions. A Nikodym set in F_q^n is a subset S so that for every point x there is a line through x whose other points lie in S; these sets are closely related to Kakeya sets and to lower bounds coming from the polynomial method (e.g., Bukh–Chao). Existing explicit small constructions are sparse outside the special case q a perfect square; this work produces new asymptotic constructions in three and higher dimensions that improve on naive/random constructions and on simple Cartesian-product approaches. Technically, AlphaEvolve discovered constructions formed by removing a controlled collection of low-degree algebraic hypersurfaces; DeepThink then suggested a generalization using Chebotarev-type heuristics and Lang–Weil counts to treat removed surfaces as quasi-random, and the author replaced some nonrigorous steps by standard concentration tools (Bennett’s inequality) to produce a rigorous intermediate bound. Attempts to push further using only quadrics were thwarted by quadratic-residue/discriminant pathology (tangency and root multiplicity), but the author salvaged a better construction by reintroducing small random subsets of those quadrics and exploiting projective symmetry and PGL-transitivity. The paper records these constructions, their probabilistic and number-theoretic analysis, and leaves as open the question of reaching the stronger heuristic bound suggested by derangement statistics for purely quadratic removals.