All Random Solved Random Open
If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that \[\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?\]
Chowla's cosine problem. The best known bound currently, due to Ruzsa [Ru04] (improving on an earlier result of Bourgain [Bo86]), replaces $N^{1/2}$ by \[\exp(O(\sqrt{\log N}).\] The example $A=B-B$, where $B$ is a Sidon set, shows that $N^{1/2}$ would be the best possible here.