Abstract
Littlewood in his 1968 monograph “Some Problems in Real and Complex Analysis” [12, Problem 22] poses the following research problem, which appears to be still open:
Problem. “If the $n_j$ are integral and all different, what is the lower bound on the number of real zeros of $\sum_{j=1}^N \cos (n_j\theta)$? Possibly $N-1$, or not much less.”
No progress seems to have been made on this in the last half century. We show that this is false.
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