The site's wording fails trivially: for $A=\{0\}$, of size $N=1$, the sum is $\cos 0=1$ for every $\theta$, and for $A=\{0,a\}$ the sum $1+\cos(a\theta)$ is never negative, so no $c>0$ and $\theta$ give a sum below $-cN^{1/2}$. The change replaces "$A\subset \mathbb{Z}$ is a finite set" by "$A\subset \mathbb{N}$ is a finite set of positive integers"; nothing else changes. The site's source [Er61, pp. 247–248] states the question in the same form, for every sequence of integers $n_1<\cdots<n_k$ with a suitable absolute constant, after Ankeny and Chowla's conjecture that the minimum tends to $-\infty$. The site's own sharpness example $A=B-B$, with $B$ a Sidon set, contains $0$ and is meant for large $N$; Bedert [Be25c] gives the same construction as the nonzero differences of $B$ (§1) and states the problem for a finite set of positive integers (abstract and Theorem 1.1). The formal-conjectures statement takes $A\subset\mathbb N$ with $0\notin A$ and all sufficiently large $N$, marked research open with no formal proof.