erdos_510

**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 $$\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?$$

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

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