{"schema":"vela.problem-packet.v0.1","problem":32,"statement":"Is there a set $A\\subset\\mathbb{N}$ such that\\[\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert = o((\\log N)^2)\\]and such that every large integer can be written as $p+a$ for some prime $p$ and $a\\in A$? Can the bound $O(\\log N)$ be achieved? Must such an $A$ satisfy\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{\\log N}> 1?\\]","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}