Green's Open Problem 39

If A ⊂ ℤ/pℤ is random, |A| = √(p), can we almost surely cover ℤ/pℤ with 100√(p) translates of A? [Gr24]

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-green-39,
  title        = {Green's Open Problem 39},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-39}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

If $A \subset Z / p Z$ is random, $∣ A ∣ = p$ , can we almost surely cover $Z / p Z$ with $100 p$ translates of $A$ ? [Gr24]

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«39». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_39 : answer(sorry) ↔
    Tendsto
      (fun p : {q : ℕ // q.Prime} ↦
        let k := Nat.sqrt p
        let c := 100 * k
        (proportionCoverable p k c : ℝ))
      atTop (𝓝 1)

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_39.variant_101 — "I do not know how to answer this even with 100 replaced by 1.01." [Gr24]"
green_39.variant_theta — Similar questions are interesting with √(p) replaced by p^θ for any θ ≤ 1/2.

References

[Gr24] Green, Ben. "100 open problems." (2024).
[BJR11] Bollobás, Béla, Svante Janson, and Oliver Riordan. "On covering by translates of a set." Random Structures & Algorithms 38.1‐2 (2011): 33-67.

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.