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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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