Chowla's cosine problem

• Slug: chowla-cosine
• List: P15
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort max
• Status: open
• Area: Harmonic analysis / additive combinatorics
• Sources: Green 100 #81; Open Problem Garden
• Started: 2026-08-17

Statement

For every n-element integer set A it is conjectured that some theta satisfies sum_{a in A} cos(2 pi a theta) <= -c sqrt(n) for an absolute c>0. The square-root negative excursion is still unproved.

Tonight

Produce a certified explicit c that is new, or a reusable auxiliary trigonometric polynomial whose bound is independently checkable and implies a strictly better universal (or n>=n0) constant than the best published one you can cite.

An SDP number with no exact certificate is not a new bound. A table of minima for a few sets is an incomplete search. Do not claim the full square-root conjecture unless you prove it.