Research log — zeros of integer cosine sums

2026-08-17

• Ben Green, 100 Open Problems, Problem 82 (Dec 2025 update), pp. 38–39. Statement: for |A|=n, how many θ ∈ ℝ/ℤ have ∑ cos(aθ) = 0? Comments record BEFL n^{5/6+o(1)}, Juškevičius–Sahasrabudhe O(n^{2/3} log^{2/3} n), Sahasrabudhe (log log log n)^{1/2−o(1)}, Bedert (log log n)^{1-o(1)}.
• Bedert, On the zeros of reciprocal Littlewood polynomials, arXiv:2312.04454 — the source listed in PROBLEM.md. This is Z_L(N) → ∞ for reciprocal ±1 polynomials, not the cosine-sum bound Green quotes as [27].
• Bedert, An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials, arXiv:2407.16075v2 (7 Jan 2025); published Israel J. Math., 30 Nov 2025. Z(N) ≥ (log log N)^{1-o(1)}. Theorem 1.3 is the precise form with unspecified c>0. Prop. 3.1 is the structured L¹ bound; Prop. 4.4 is the period/K structure; §5 is the one-page combination.
• Erdélyi, Improved lower bound for the number of unimodular zeros…, arXiv:1702.05823 / Acta Arith. 192 (2020). Theorem 1.7: Littlewood with constant 1/30 on [0,2π]. Lemmas 3.7–3.8: explicit m, d_m < 3^m, log q. Theorem 2.1 is the weaker (log log log |P(1)|)/(log log log log |P(1)|) bound.
• Sahasrabudhe, Counting zeros of cosine polynomials, arXiv:1610.07680v3 / Adv. Math. 343 (2019). Lemmas 10–11: windows, periods ≤ 16 t log log(t+3), φ(n) ≥ n/(8 log log n) for n>3.
• Juškevičius–Sahasrabudhe, Cosine polynomials with few zeros, arXiv:2005.01695v3 / Bull. Lond. Math. Soc. 53 (2021). Construction f = D_n − g, 𝔼 Z = Θ(n log m / √m + m), optimum O((n log n)^{2/3}). Deterministic Lemma 3.1: Z vs |E(g)|.
• Konyagin, On zeros of sums of cosines, Math. Notes 108 (2020) — independent O((N log N)^{2/3}).
• Borwein–Erdélyi–Ferguson–Lockhart, Ann. of Math. 167 (2008) — original O(N^{5/6} log N) counterexample to Littlewood's N−1.
• McGehee–Pigno–Smith, Ann. of Math. 113 (1981); DeVore–Lorentz book proof (Erdélyi's 1/30). Stegeman, Math. Ann. 261 (1982) has 4/π³ for unweighted exponential sums; we used Erdélyi's 1/30 because it includes general coefficients ∑ |a_j|/j.
• MathOverflow 312265 — Stegeman / Bloom comments on the Littlewood L¹ constant.
• No source found tonight states a numerical c in Z(N) ≥ c log log N / log log log N, nor a construction beating O((N log N)^{2/3}). Later 2025–26 papers touching cosine polynomials (Erdélyi arXiv:2607.21877, Stankov, Brazitikos arXiv:2605.18087) do not improve Z(N).

Adjacent work tonight

• problems/chowla-cosine extracted K(n) ≥ n^{1/7}/18 from a different Bedert paper (arXiv:2509.05260). Not reused except as a style template for an explicit-constant certificate. That folder was not modified.