{"schema":"vela.problem-packet.v0.1","problem":741,"statement":"Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive (upper) density. Can one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive (upper) density?Is there a basis $A$ of order $2$ such that if $A=A_1\\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[{"verdict":"faithful","attestedBy":"reviewer:will-blair","formalRef":"erdos_741.parts.i.lean","targetFinding":"vf_748cf8097c3816cf"}],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}