Erdős Problem 510

Reference: erdosproblems.com/510

theorem Erdos510.erdos_510 : (∃ (c : ℝ) (_ : 0 < c), ∀ (N : ℕ) (A : Finset ℤ), A.card = N → ∃ (θ : ℝ), ∑ n ∈ A, Real.cos (↑n * θ) < -c * √↑N) ↔ sorry

Chowla's cosine problem

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