Polynomial bounds for the Chowla Cosine Problem

Abstract

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 $$\underset{x}{\min }{f}_A(x)⩽-n^{1/5-o(1)},$$ thereby establishing polynomial bounds for the Chowla cosine problem.

21 pages. New version improves the exponent to 1/5