Thin cyclic additive bases

• Slug: thin-cyclic-bases
• List: P09
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
• Status: open
• Area: Additive combinatorics / design theory
• Sources: Green 100 #33
• Started: 2026-08-17

Statement

It is unknown whether infinitely many q admit A ⊆ Z/qZ with A+A = Z/qZ and |A| = (√2+o(1))√q, the counting-optimal constant.

Tonight

An infinite parametric family, or a certified finite construction beating the published density for some q, with an independent verifier. Isolated small-q tables are an incomplete search unless they imply an infinite family. Fetch Green #33 before searching.

Record after 2026-08-17

Green #33 remains open. Replayed Bevan–Erskine–Lewis: infinitely many $n=6q(q-2)$ with $\mathrm{SS}(n,2)=(4+o(1))q$, ratio $\sqrt{8/3}+o(1)$. No family with a smaller liminf was found. See ATTACK.md, WALKTHROUGH.md, RESEARCH.md.