On the zeros of cosine polynomials: solution to a problem of Littlewood
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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 Σ j=1N cos(n j Θ)? 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. THEOREM. There exists a cosine polynomial Σ j=1N cos(n j Θ) with the n j integral and all different so that the number of its real zeros in the period [-π, π) is O (N 5/6 log N).