Abstract
It is shown that there exist arbitrarily large natural numbers \(N\) and distinct nonnegative integers \(n_1,\dots,n_N\) for which the number of zeros on \([-\pi,\pi)\) of the trigonometric polynomial \(\sum_{j=1}^N \cos(n_j t)\) is \(O(N^{2/3}\log^{2/3} N)\).
Similar content being viewed by others
References
J. E. Littlewood, Some Problems in Real and Complex Analysis (D. C. Heath and Co., Lexington, MA, 1968).
P. Borwein, T. Erdélyi, R. Ferguson, and R. Lockhart, “On the zeros of cosine polynomials: solution to a problem of Littlewood,” Ann. of Math. (2) 167 (3), 1109–1117 (2008).
T. Erdélyi, “The number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set,” Acta Arith. 176 (2), 177–200 (2016).
T. Erdélyi, “Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set,” Acta Arith. 192 (2), 189–210 (2020).
J. Sahasrabudhe, “Counting zeros of cosine polynomials: on a problem of Littlewood,” Adv. Math. 343 (5), 495–521 (2019).
T. Ju\^skevi\^cius and J. Sahasrabudhe, Cosine Polynomials with Few Zeros, arXiv: 2005.01695v1 (2020).
Acknowledgments
The author wishes to express gratitude to K. S. Ryutin for his interest in the paper and useful remarks. Without his careful reading, the paper stood little chance of being sent to press.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Konyagin, S.V. On Zeros of Sums of Cosines. Math Notes 108, 538–541 (2020). https://doi.org/10.1134/S0001434620090254
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434620090254