Problem 510

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claims/: The 0 claim pages of Problem 510, one per claimant's result; the problem's standing derives from them.

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Statement. If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that

$$\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?$$

Statement (corrected). If $A\subset \mathbb{N}$ is a finite set of positive integers of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that

$$\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?$$

Notes. The site's wording fails trivially: for $A=\{0\}$, of size $N=1$, the sum is $\cos 0=1$ for every $\theta$, and for $A=\{0,a\}$ the sum $1+\cos(a\theta)$ is never negative, so no $c>0$ and $\theta$ give a sum below $-cN^{1/2}$. The change replaces "$A\subset \mathbb{Z}$ is a finite set" by "$A\subset \mathbb{N}$ is a finite set of positive integers"; nothing else changes. The site's source [Er61, pp. 247–248] states the question in the same form, for every sequence of integers $n_1<\cdots<n_k$ with a suitable absolute constant, after Ankeny and Chowla's conjecture that the minimum tends to $-\infty$. The site's own sharpness example $A=B-B$, with $B$ a Sidon set, contains $0$ and is meant for large $N$; Bedert [Be25c] gives the same construction as the nonzero differences of $B$ (§1) and states the problem for a finite set of positive integers (abstract and Theorem 1.1). The formal-conjectures statement takes $A\subset\mathbb N$ with $0\notin A$ and all sufficiently large $N$, marked research open with no formal proof.

Status. Open, the site's label (page last edited 28 September 2025), which fits the corrected Statement.

Source. erdosproblems.com/510, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #510, https://www.erdosproblems.com/510.

References.

• [Be25c] B. Bedert, Polynomial bounds for the Chowla Cosine Problem. arXiv:2509.05260 (2025).
• [Bo86] Bourgain, J., Sur le minimum d'une somme de cosinus. Acta Arith. 45 (1986), 381-389.
• [Er61] Erdős, P., Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221–254, pp. 247–248.
• [JMTZ25] Z. Jin, A. Milojević, I. Tomon, and S. Zhang, From small eigenvalues to large cuts, and Chowla's cosine problem. arXiv:2509.03490 (2025).
• [Ru04] Ruzsa, Imre Z., Negative values of cosine sums. Acta Arith. (2004), 179-186.

Formalization. Statement in formal-conjectures.

Current assessment

The question (site formulation). Chowla's cosine problem: whether an absolute $c>0$ exists such that every finite $A\subset\mathbb Z$ of size $N$ has some $\theta$ with $\sum_{n\in A}\cos(n\theta)<-cN^{1/2}$. The site labels the problem OPEN (page last edited 28 September 2025); the site's wording fails at the sets $\{0\}$ and $\{0,a\}$, as the Notes above record, and the corrected Statement for sets of positive integers is the question the literature studies.

Standing. The corrected Statement is open. For a set $A$ of $N$ positive integers write $m(A)=\min_\theta\sum_{n\in A}\cos(n\theta)$. Bourgain [Bo86] proved $m(A)\le-\exp((\log N)^{\varepsilon})$ for an absolute $\varepsilon>0$, and Ruzsa [Ru04] improved this to $m(A)\le-\exp(c\sqrt{\log N})$ for an absolute $c>0$. Polynomial bounds were proved independently in September 2025: Jin, Milojević, Tomon and Zhang [JMTZ25] obtain $m(A)\le-N^{1/10-o(1)}$ from a spectral theorem on graphs with small least eigenvalue (card), and Bedert [Be25c] obtains $m(A)\le-N^{1/12}$ by a five-page argument in the first arXiv version and $m(A)\le-cN^{1/7}$ in the second, of 23 September 2025 (card). The best bound is Bedert's $m(A)\le-N^{1/5-o(1)}$, Theorem 1.1 of the third arXiv version of 24 July 2026, announced in a thread post that day; the site's commentary, last edited 28 September 2025, gives the second version's $-cN^{1/7}$. The example $A=B-B$ with $B$ a Sidon set shows that $N^{1/2}$ would be best possible. A thread post of 26 August 2026 tabulates certified upper bounds on the extremal value $-\sup_A m(A)$ over sets of $n$ positive integers for $4\le n\le17$; it is not a dated manuscript and settles no instance. None of these bounds settles an instance of the corrected question, so none is a claim, and the problem has no claim page.

Search scope. The site's page, its three comments and its empty proof-claims tab, the arXiv records of [Be25c] and [JMTZ25], the formal-conjectures statement file and the library cards of [Bo86], [Be25c] and [JMTZ25]; [Ru04] and [Er61] are cited from the site and the Rényi archive scan.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• kolountzakis_1996_density_b_h_g_sequences_minimum
• kolountzakis_1996_density_b_h_g_sequences_minimum / theorem_2
• bedert_2025_polynomial_bounds_chowla_cosine_problem
• bourgain_1986_sur_le_minimum_d_une_somme
• jin_2025_small_eigenvalues_large_cuts_chowla_s
• konyagin_1981_littlewood_problem
• konyagin_1981_littlewood_problem / corollary_3