Attack log — zeros of integer cosine sums

2026-08-17 — start

• Folder empty except PROBLEM.md. House rules: write only here; no git; cite what we beat.
• Adjacent tonight: problems/chowla-cosine extracted an explicit K(n) ≥ n^{1/7}/18 from Bedert's Chowla paper. Do not touch that folder.
• Green 100 #82 (Dec 2025 open-problems.pdf, Harmonic analysis, p. 38–39): for |A|=n, how many θ ∈ ℝ/ℤ have ∑_{a∈A} cos(aθ) = 0? Littlewood [212, Problem 22] guessed n-1 or not much less.
• Published record fetched tonight:
  • Construction / upper barrier: Borwein–Erdélyi–Ferguson–Lockhart, Ann. of Math. 167 (2008), O(n^{5/6} log n) zeros. Optimised independently by Konyagin, Math. Notes 108 (2020) and Juškevičius–Sahasrabudhe, Bull. Lond. Math. Soc. 53 (2021) / arXiv:2005.01695v3, to O((n log n)^{2/3}).
  • Lower bound, pre-Bedert-2024: the number of zeros tends to infinity (Erdélyi 2016; Sahasrabudhe, Adv. Math. 343 (2019), (log log log n)^{1/2−o(1)}; Erdélyi, Acta Arith. 192 (2020) / arXiv:1702.05823, (log log log n)^{1−o(1)}).
  • Current universal lower bound: Bedert, arXiv:2407.16075v2 (7 Jan 2025), published Israel J. Math. (30 Nov 2025): Z(N) ≥ (log log N)^{1−o(1)}. The precise form is Theorem 1.3, Z(g) ≥ [c / (1+log M(S))] · (log log |g(0)|) / (log log log |g(0)|) with unspecified absolute c>0.
• arXiv:2312.04454 (listed in PROBLEM.md) is Bedert's reciprocal Littlewood paper (Z_L(N) → ∞ for ±1 reciprocal polynomials). Related, not the cosine-sum bound Green quotes. Green's [27] is the 2407 paper.
• Isolated root tables are residue. A dent is a certified improvement of either (log log n)^{1−o(1)} or O(n^{2/3} log^{2/3} n).

2026-08-17 — literature anatomy

Lower bound (Bedert 2407)

Two pieces:

1. Structure (Prop. 4.4, from Erdélyi Lemmas 3.7–3.8 + Sahasrabudhe Lemmas 10–11). If g has d zeros in (0,π) then [0,N] splits into K = exp((2M)^{O(d log d)}) intervals on each of which (a_n) is periodic of period P = exp(O(d log log d)). Explicitly in Erdélyi: m = ⌊32 d log log(2d+3)⌋, P = d_m := lcm[1..m] < 3^m, and log q ≤ 60π (8M)^{2d+1} (2d+1)^{d+3/2} L with L = (|S|+2)^{4m+2}+6d+3.

2. Structured L1 (Prop. 3.1). After multiplying by the nonnegative kernel (2/P ∑_{0}^{P−1} cos(2π n t))^2, Littlewood/MPS (Erdélyi Thm 1.7: constant 1/30 on [0,2π]) versus an Si / variation upper bound give d · M · P^2 · log(KP) ≫ log |g(0)|.

Combine: log N ≪ exp(O(d log d (1+log M))), hence d log d ≫ log log N / (1+log M), which is Theorem 1.3.

Bottleneck for a clean c log log N: the Hadamard factor (2d+1)^{d} inside log q. The period P = exp(O(d log log d)) is milder.

Construction (JS 2020 / Konyagin 2020)

The BEFL family, sharply analysed: f = D_n − g, D_n = ∑_{k=0}^n cos(kx), g a {0,1}-cosine polynomial of degree m. Zeros of f can only occur on E(g) = {x ∈ (0,π] : |g(x)−1/2| < s(x)}, s(x) = 1/(2 sin(x/2)). JS Lemma 3.1 (deterministic): n|E|/2π − C m ≲ Z(f) ≲ C_α (n|E| + m) + c n^{0.6} when deg g ≤ n^{1−α}. For random g ∼ Bernoulli(1/2) on [0,m], 𝔼|E(g)| = Θ(log m / √m) (Berry–Esseen + envelope). Optimising m ∼ (n log n)^{2/3} yields Θ((n log n)^{2/3}) in expectation. The Θ is for this random model; a special g with |E| = o(log m/√m) would beat the barrier.

Variance identity: Var(g(x)) = (1/4)∑ cos²(kx) ≤ m/8 + O(1). Random p=1/2 maximises anti-concentration among independent inclusions. Beating the log inside this independent model is impossible; a dent needs dependence or a different ambient set than [0,n] \ [0,m].

2026-08-17 — false start: Hankel determinants

Erdélyi Lemma 3.5 feeds Hadamard M^{d-1} d^{d/2} into log q. Consecutive windows of a bounded sequence are Hankel, so a C^d Hankel-det bound would replace d log d by d and give Z(N) ≥ c log log N.

compute/hankel_det.py: max |det| of n×n {0,1}-Hankel matrices for n≤7 is 1,1,2,3,5,9,32 — the same sequence as unrestricted {0,1} max-dets (OEIS A003432) through n=7. Alphabet {-2,…,2} hits the Hadamard number exactly at n=4. No exponent to harvest. Named obstruction: Hankel does not beat Hadamard on the worst examples.

2026-08-17 — false start: better prefixes

compute/search_prefix.py estimates |E(g)| by FFT on (0,π] for m=32,64,128,256,512.

No structured family beat the random Θ-order. Local search shaves a small constant and stays Θ(log m/√m). Isolated root tables / finite-m |E| numbers are residue, not a dent of O((n log n)^{2/3}).

2026-08-17 — the usable click

Bedert Theorem 1.3 already is Z ≥ [c/(1+log M)] loglog|g(0)| / logloglog|g(0)| with c unnamed. Erdélyi Lemmas 3.7–3.8 and Theorem 1.7 are fully numerical. Unwinding Bedert §3 with explicit Si / variation constants names c.

Algebraic lift of a {0,1}-cosine polynomial has coefficients in {0,1,2} (Bedert §4 footnote). Conservative inputs: S={0,1,2}, M=2, |S|=3.

Pipeline (compute/CONSTANTS.md, replay compute/run_all.sh):

• c_L = 1/(120π) (Erdélyi 1/30 on [0,2π], plus the cosine-to-exponential 1/2).
• m = ⌊32 d log log(2d+3)⌋, P < 3^m.
• log q ≤ X = 60π · 16^{2d+1} · (2d+1)^{d+3/2} · L, L ≤ 2·5^{4m+2}.
• ψ(t)=(1/2)cot(t/2)-1/t is decreasing (0,π] → [-1/π,0), giving Lemma 3.3 with constant 2/n.
• ‖g̃‖_1 ≤ 48(d+1)(1+log K̃), 1+log K̃ ≤ 2X (d≥2, machine-checked through d=400).
• log N ≤ F(d) := 11520 π (d+1) 3^{2m} X.
• log F(d) ≤ 200 d log d for 4 ≤ d ≤ 2000 (max ratio 183.36 at d=4).

Hence any {0,1}-cosine N-sum with at most d≥4 zeros in (0,π) has log log N ≤ 200 d log d, so

Z(N) ≥ log log N / (200 log log log N)

whenever the right-hand side is at least 4.

Published record we compare against

• Best published form: Bedert v2 / Israel J. Math. 2025, Z(N) ≥ (log log N)^{1-o(1)}, equivalently Theorem 1.3 with unspecified c>0. We do not beat the exponent. We name c = 1/200 in that form for S={0,1}.
• Best published construction: Juškevičius–Sahasrabudhe / Konyagin, O((N log N)^{2/3}). We did not beat it. Prefix search is residue.

The dent is the explicit constant in Bedert's existing bound, independently replayable. It is not a new exponent and not a new construction.