Research log — Chowla's cosine problem

2026-08-17

• Ben Green, 100 Open Problems, Problem 81 (Dec 2025 update). Statement: ∑_{a∈A} cos(aθ) ≤ −c √n? Comments record Ruzsa exp(−c √log n) as the pre-2025 record and Bedert / JMTZ as the 2025 polynomial bounds, quoting Bedert as any exponent < 1/7 (that is the v2 exponent; v3 is 1/5−o(1)).
• Open Problem Garden, Chowla's cosine problem.
• Bedert, Polynomial bounds for the Chowla Cosine Problem, arXiv:2509.05260v3 (24 Jul 2026). K(n) ≫ n^{1/5−o(1)}; §5 gives n^{1/12}; §7 gives n^{1/7} then the log-improved |B_t|. No numerical c. Lemma 7.2 is the 32-window table we re-checked.
• Jin–Milojević–Tomon–Zhang, From small eigenvalues to large cuts, and Chowla's cosine problem, arXiv:2509.03490v2. n^{1/10−o(1)}. Spectral-graph method; no numerical c.
• Ruzsa, Negative values of cosine sums, Acta Arith. 111 (2004). K(n) ≥ exp(c' (log n)^{1/2}). Lemma 4.2 / 3.1 is the AP bound we instantiate as L ≤ 8K^2+8.
• Roth, On cosine polynomials corresponding to sets of integers, Acta Arith. 24 (1973). Energy lemma (Bedert Lemma 4.1).
• McGehee–Pigno–Smith, Hardy's inequality and the L¹-norm of exponential sums, Ann. of Math. 113 (1981); Konyagin, Izv. Akad. Nauk SSSR 45 (1981). L¹ ≫ log n, hence K(n) ≫ log n. Steen notes: A = 128 is an admissible L¹ constant.
• Brazitikos, Sharp estimates for conjugate functions, arXiv:2605.18087 (18 May 2026). Cites Bedert n^{1/7−o(1)} and JMTZ n^{1/10−o(1)}; no better Chowla exponent and no explicit c.
• Quanta, 28 Jan 2026 — popular account of the 2025 polynomial bounds (quotes Bedert n^{1/7} as of that date).
• Sidon-difference construction (Bedert §1, following the classical example): K(n) ≪ √n. Chowla 1965 conjectured this is sharp.

No source found tonight states an explicit numerical c in K(n) ≥ c n^α for a fixed α > 0.