OP-01043 Partially resolved

Chowla's cosine problem

Let $A$ be a set of $n$ positive integers. Ruzsa showed that the sum can always be made at most $-e^{c\sqrt{\log n}}$, and Bedert and, independently, Jin, Milojević, Tomon and Zhang obtained $-n^{c}$ for some $c > 0$.

Problem. Is there a $\theta$ with $\sum_{a \in A} \cos(a\theta) \le -c\sqrt{n}$, for an absolute constant $c > 0$?

Source: S. Chowla, Some applications of a method of A. Selberg, J. Reine Angew. Math. 217 (1965), from a question of N. C. Ankeny and S. Chowla (1948); Erdős problem #510. Listed as Problem 81 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: Chowla's cosine problem, cosine polynomials, minimum of trigonometric sums

References

@article{chowla1965selberg,
  author = {Chowla, S.},
  title = {Some applications of a method of {A}. {S}elberg},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume = {217},
  pages = {128--132},
  year = {1965},
  doi = {10.1515/crll.1965.217.128}
}

@article{ruzsa2004cosine,
  author = {Ruzsa, Imre Z.},
  title = {Negative values of cosine sums},
  journal = {Acta Arithmetica},
  volume = {111},
  number = {2},
  pages = {179--186},
  year = {2004},
  doi = {10.4064/aa111-2-6}
}

@misc{jin2025chowla,
  author = {Jin, Zhihan and Milojevi{\'c}, Aleksa and Tomon, Istv{\'a}n and Zhang, Shengtong},
  title = {From small eigenvalues to large cuts, and {C}howla's cosine problem},
  year = {2025},
  eprint = {2509.03490},
  archivePrefix = {arXiv}
}

@misc{bedert2025chowla,
  author = {Bedert, Benjamin},
  title = {Polynomial bounds for the {C}howla cosine problem},
  year = {2025},
  eprint = {2509.05260},
  archivePrefix = {arXiv},
  note = {Version 3 (2026) improves the exponent to 1/5}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 81; version of January 2026, most recent update December 2025},
  year = {2026}
}

Resolution claims

Polynomial bounds for the Chowla Cosine Problem v1 arXiv ↗ AI-Verified

Benjamin Bedert

Let $A\subset \mathbf{N}$ be a finite set of $n=|A|$ positive integers, and consider the cosine sum $f_A(x)=\sum_{a\in A}\cos ax$. We prove that $$\min_x f_A(x)\leqslant -n^{ 1/5-o(1)},$$ thereby establishing polynomial bounds for the Chowla cosine problem.

Partial resolution · not verified · fidelity not established · posted on arXiv 24 Jul 2026

Status history

When	Change	By	Reason
2026-10-29	Open → Partially resolved	The registry's curators	Polynomial bounds for the Chowla Cosine Problem v1 recorded from arXiv:2509.05260v3.

Citing this problem

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

OP-01043, "Chowla's cosine problem", clopen.solutions.