Chowla’s square-root conjecture for cosine sums

Establish that the extremal quantity \(\mathcal K(N)=\inf_{S\subset\mathbb N,\,|S|=N}K(S)\), where \(K(S)=-\min_{x\in\mathbb R/\mathbb Z}\sum_{s\in S}\cos(2\pi sx)\), has optimal order of magnitude \(\sqrt N\); equivalently, prove the lower bound \(\mathcal K(N)\gg \sqrt N\).

Background

Chowla’s cosine problem concerns the smallest possible negative minimum of a cosine sum associated with an $N$-element set of positive integers. The paper proves the polynomial lower bound $\mathcal K(N)\gg N^{1/5}$, improving the previously known $N^{1/5-o(1)}$ estimate by removing a logarithmic loss.

The paper notes that a Sidon-difference construction gives the matching-order upper bound $\mathcal K(N)\ll\sqrt N$. Thus the unresolved issue is to improve the lower bound from exponent $1/5$ to the conjectured exponent $1/2$, which would determine the optimal order of magnitude.