Polynomial bounds for the Chowla Cosine Problem

Let $A\subset \mathbf{N}$ be a finite set of $n=|A|$ positive integers, and consider the cosine sum $f_A(x)=\sum_{a\in A}\cos ax$.

We prove that $$\min_x f_A(x)\leqslant -n^{ 1/5-o(1)},$$ thereby establishing polynomial bounds for the Chowla cosine problem.