Ben Green's Open Problem 82

Let A ⊂ ℤ be a set of size n. For how many θ ∈ ℝ/ℤ must we have Σ_a ∈ A cos(2π aθ) = 0? The answer is the function minZeros.

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-82,
  title        = {Ben Green's Open Problem 82},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-82}},
  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

Let $A \subset Z$ be a set of size $n$ . For how many $θ \in R / Z$ must we have $\sum_{a \in A} cos (2 π a θ) = 0$ ? The answer is the function minZeros.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«82». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_82 : (answer(sorry) : ℕ+ → ℕ∞) = minZeros

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.

References

Ben Green's Open Problem 82
An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials (Bedert, 2025)
Cosine polynomials with few zeros (Juškevičius & Sahasrabudhe, 2020)

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.