Ben Green's Open Problem 68

The question

Suppose that $A\subset {\mathbb{F}}_p^2$ is a set meeting every line in at most 2 points. Is it true that all except $o(p)$ points of $A$ lie on a cubic curve?

Such sets are called arcs. We read "cubic curve" as the zero set of a nonzero polynomial of total degree at most $3$ in two variables over ${\mathbb{F}}_p$. This includes conics and lines, and unions of these. The $o(p)$ is uniform in $A$: for every $\epsilon >0$ and every large prime $p$, every arc $A$ has all but at most $\epsilon p$ of its points on one such curve. Green asked a projective form of this question in [Gr14, Section 4.4]. [Gr24] notes that the question is more natural in projective space, and that the analogue over ${\mathbb{F}}_q$ with $q=p^2$ is false, because of Hermitian quadrics.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«68».

theorem green_68 :
    ∀ ε : ℝ, 0 < ε → ∃ p₀ : ℕ, ∀ (p : ℕ) [Fact p.Prime], p₀ ≤ p →
      ∀ A : Finset (Fin 2 → ZMod p), MeetsLinesInAtMost 2 A →
        ∃ F : MvPolynomial (Fin 2) (ZMod p), F ≠ 0 ∧ F.totalDegree ≤ 3 ∧
          (numOffCurve F A : ℝ) ≤ ε * p

What counts as progress

• A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

• green_68.variants.bounded_intersection — More generally than in green_68, one may conjecture that if A ⊂ 𝔽_p^2 meets every line in O(1) points, then all but o(p) points of A lie on a curve of degree…
• green_68.variants.f_four_little_o — Denote by f_d(p) the size of the largest no-three-in-a-line set in 𝔽_p^d. Is f_4(p) = o(p^3)?
• green_68.variants.f_four_polynomial — Is f_4(p) = O(p^2+o(1)), where f_d(p) is the size of the largest no-three-in-a-line set in 𝔽_p^d?
• green_68.variants.blocking_set — Let PG(2, q) be the projective plane over 𝔽_q. Is there a set B ⊂ PG(2, q) (a "blocking set") meeting every line in at least 1, but at most 1000, points?

References

• [Gr24] Green, Ben. "100 open problems." (2024).
• [Gr14] Green, B., Approximate algebraic structure. Proceedings of the ICM 2014, Vol. I, Kyung Moon Sa, Seoul (2014), 341--367. arXiv:1404.0093
• [Vo90] Voloch, J. F., Arcs in projective planes over prime fields. J. Geom. 38 (1990), 198--200.
• [BaLa19] Ball, S. and Lavrauw, M., Arcs in finite projective spaces. EMS Surv. Math. Sci. 6 (2019), 133--172. arXiv:1908.10772

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.