Polynomial bounds for the Chowla Cosine Problem

Let $A\subset \mathbf{N}$ be a finite set of $n=\mathrm{\mid }A\mathrm{\mid }$ positive integers, and consider the cosine sum $f_A(x)={\sum }_{a\in A}\cos ax$. We prove that $$\underset{x}{\min }{f}_A(x)⩽-n^{1\mathrm{/}7-o(1)},$$ thereby establishing polynomial bounds for the Chowla cosine problem.