{"schema":"vela.problem-packet.v0.1","problem":38,"statement":"Does there exist $B\\subset\\mathbb{N}$ which is not an additive basis, but is such that for every set $A\\subseteq\\mathbb{N}$ of Schnirelmann density $\\alpha$ and every $N$ there exists $b\\in B$ such that\\[\\lvert (A\\cup (A+b))\\cap \\{1,\\ldots,N\\}\\rvert\\geq (\\alpha+f(\\alpha))N\\]where $f(\\alpha)>0$ for $0<\\alpha <1 $?The Schnirelmann density is defined by\\[d_s(A) = \\inf_{N\\geq 1}\\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N}.\\]","status":"solved","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)"}