OP-00991 Open

Zeros of cosine polynomials with 0-1 coefficients

Let $A \subset \mathbb{Z}$ have size $n$. Littlewood suggested "probably $n - 1$, or not much less" for the question below. Borwein, Erdélyi, Ferguson and Lockhart gave examples with at most $n^{5/6 + o(1)}$ zeros, and Juškevičius and Sahasrabudhe, and independently Konyagin, improved this to $O(n^{2/3} \log^{2/3} n)$. Erdélyi and Sahasrabudhe showed that the minimum number of zeros grows without bound, Sahasrabudhe at the rate $(\log\log\log n)^{1/2 - o(1)}$; Bedert improved this to $(\log\log n)^{1 - o(1)}$.

Problem. For how many $\theta \in \mathbb{R}/\mathbb{Z}$ must $\sum_{a \in A} \cos(2\pi a\theta)$ vanish?

Source: J. E. Littlewood, Some problems in real and complex analysis (1968), Problem 22. Listed as Problem 82 in B. Green, 100 open problems (version of January 2026).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: CA — Classical Analysis and ODEs
Secondary: CO, NT
Keywords: cosine polynomials, Littlewood's problems, zeros of trigonometric polynomials
Literature checked: Oct. 26, 2026 by clemens-admin

References

@book{littlewood1968,
  author = {Littlewood, J. E.},
  title = {Some Problems in Real and Complex Analysis},
  series = {Heath Mathematical Monographs},
  publisher = {D. C. Heath},
  year = {1968},
  note = {Problem 22}
}

@article{borwein2008zeros,
  author = {Borwein, Peter and Erd{\'e}lyi, Tam{\'a}s and Ferguson, Ronald and Lockhart, Richard},
  title = {On the zeros of cosine polynomials: solution to a problem of {L}ittlewood},
  journal = {Annals of Mathematics (2)},
  volume = {167},
  number = {3},
  pages = {1109--1117},
  year = {2008},
  doi = {10.4007/annals.2008.167.1109}
}

@article{juskevicius2021sahasrabudhe,
  author = {Ju{\v{s}}kevi{\v{c}}ius, Tomas and Sahasrabudhe, Julian},
  title = {Cosine polynomials with few zeros},
  journal = {Bulletin of the London Mathematical Society},
  volume = {53},
  number = {3},
  pages = {877--892},
  year = {2021},
  doi = {10.1112/blms.12468}
}

@article{konyagin2020zeros,
  author = {Konyagin, S. V.},
  title = {On zeros of sums of cosines},
  journal = {Mathematical Notes},
  volume = {108},
  number = {3-4},
  pages = {538--541},
  year = {2020},
  doi = {10.1134/S0001434620090254}
}

@article{erdelyi2016unimodular,
  author = {Erd{\'e}lyi, Tam{\'a}s},
  title = {The number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set},
  journal = {Acta Arithmetica},
  volume = {176},
  number = {2},
  pages = {177--200},
  year = {2016},
  doi = {10.4064/aa8442-7-2016}
}

@article{sahasrabudhe2019,
  author = {Sahasrabudhe, Julian},
  title = {Counting zeros of cosine polynomials: on a problem of {L}ittlewood},
  journal = {Advances in Mathematics},
  volume = {343},
  pages = {495--521},
  year = {2019},
  doi = {10.1016/j.aim.2018.11.025}
}

@article{bedert2025littlewood,
  author = {Bedert, Benjamin},
  title = {An improved lower bound for a problem of {L}ittlewood on the zeros of cosine polynomials},
  journal = {Israel Journal of Mathematics},
  volume = {274},
  number = {1},
  pages = {483--500},
  year = {2025},
  doi = {10.1007/s11856-025-2872-5},
  eprint = {2407.16075},
  archivePrefix = {arXiv}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 82; version of January 2026, most recent update December 2025},
  year = {2026}
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00991, "Zeros of cosine polynomials with 0-1 coefficients", clopen.solutions.