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On the zeros of cosine polynomials: solution to a problem of Littlewood. (English) Zbl 1186.11045
The problem in the title states: “If the \(n_j\) are distinct integers, what is the lower bound on the number of real zeros of \(\sum^N_{j=1} \cos(n_j\theta)\)? Possibly \(N-1\), or not much less.” The authors prove the following theorem, which shows that \(N-1\) is not correct. There exists such a cosine polynomial whose number of zeros in the interval \([-\pi, \pi)\) is \(O(N^{5/6}\log N)\).

MSC:
11L03 Trigonometric and exponential sums (general theory)
42A05 Trigonometric polynomials, inequalities, extremal problems
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