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arXiv · 2407.16075

An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials

Benjamin Bedert
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Abstract

Let $Z(N)$ denote the minimum number of zeros in $[0,2\pi]$ that a cosine polynomial of the form $$f_A(t)=\sum_{n\in A}\cos nt$$ 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 $Z(N)\geqslant (\log\log N)^{(1+o(1))}$ which exponentially improves on the previous best bounds of the form $Z(N)\geqslant (\log\log\log N)^c$ due to Erd\'elyi and Sahasrabudhe.

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Benjamin Bedert. 2024-07-22. An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials. https://arxiv.org/abs/2407.16075

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