Erdős Problem #510 — open problem in number theory

Erdős Problem #510

Chowla's cosine problem If A⊂ ℕ is a finite set of positive integers of size N > 0 then is there some absolute constant c>0 and θ such that Σ_n∈ Acos(nθ) < -cN^1/2?

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-erdos-510,
  title        = {Erdős Problem #510},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-510}},
  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

Chowla's cosine problem

If $A \subset N$ is a finite set of positive integers of size $N > 0$ then is there some absolute constant $c > 0$ and $θ$ such that $n \in A \sum cos (n θ) < - c N^{1/2} ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«510». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_510 :
    answer(sorry) ↔ ∃ (c : ℝ) (hc : 0 < c),
      ∀ᶠ N in atTop, ∀ (A : Finset ℕ), 0 ∉ A → #A = N →
      ∃ θ, ∑ n ∈ A, cos (n * θ) < -c * sqrt N

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 — check erdosproblems.com/510. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/510
Ben Green's Open Problem 81
[Ru04] Ruzsa, Imre Z., Negative values of cosine sums. Acta Arith. (2004), 179-186.
[Be25c] B. Bedert, Polynomial bounds for the Chowla Cosine Problem. arXiv:2509.05260 (2025).

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.