An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials

Let $Z(N)$ denote the minimum number of zeros in $[0,2\pi ]$ that a cosine polynomial of the form $$f_A(t)=\sum _{n\in A}\cos nt$$ can have when $A$ is a finite set of non-negative integers of size $\mathrm{\mid }A\mathrm{\mid }=N$. It is an old problem of Littlewood to determine $Z(N)$. In this paper, we obtain the lower bound $Z(N)⩾(\log \log N{)}^{(1+o(1))}$ which exponentially improves on the previous best bounds of the form $Z(N)⩾(\log \log \log N{)}^c$ due to Erdélyi and Sahasrabudhe.