Zeros of integer cosine sums

• Slug: cosine-zeros
• List: P16
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
• Status: open
• Area: Harmonic analysis
• Sources: Green 100 #82; Bedert, arXiv:2312.04454
• Started: 2026-08-17

Statement

For |A|=n, the best universal lower bound on the number of zeros of sum_{a in A} cos(2 pi a theta) grows only like (log log n)^{1-o(1)}, while examples give only an O(n^{2/3} log^{2/3} n) upper barrier. The true order is unknown.

Tonight

A certified improvement of either the universal lower bound or the construction-side upper barrier, with an independent verifier. Isolated root tables for a few sets are an incomplete search. Fetch Bedert and Green #82 before searching. Do not claim a full determination of the order.

Outcome (2026-08-17)

Named the constant in Bedert Theorem 1.3 for {0,1}-cosine sums: Z(N) ≥ log log N / (200 log log log N) whenever the right-hand side is at least 4. Replay: compute/run_all.sh. Did not beat the exponent (log log N)^{1-o(1)} and did not beat the construction O((N log N)^{2/3}). Prefix search and Hankel-det experiment are an incomplete search.