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 π ] that a cosine polynomial of the form fA(t)=∑n ∈ A cos nt f A ( t ) = ∑ n ∈ A cos n t can have when A is a finite set of non-negative integers of size ∣ A ∣ = 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.