Source labels openChecked July 26, 2026

Erdős Problem 510

Chowla's cosine problem*

Mathematical statement

Chowla's cosine problem*

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

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_510

Canonical source

Complete statement target, proof intentionally absentLean 4

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 := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization erdosproblems.com/510background Ben Green's Open Problem 81background