{"schema":"vela.problem-packet.v0.1","problem":33,"statement":"Let $A\\subset\\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\\in A$ and $n\\geq 0$. What is the smallest possible value of\\[\\limsup \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}?\\]Is\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>1?\\]","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_658a4e5e0f56bbcb","kind":"partial_proof","claim":"Erdős 33 (smallest limsup |A∩[1,N]|/√N for additive complements of squares) is open; exact value unknown. I confirmed the record sandwich inf ∈ [4/π, 2φ^{5/2}]≈[1.273, 6.660], gave a clean self-contai","grade":"partial_proof","gateStatus":"needs_verification","superseded":false}],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}