{"schema":"vela.problem-packet.v0.1","problem":354,"statement":"Let $\\alpha,\\beta\\in \\mathbb{R}_{>0}$ such that $\\alpha/\\beta$ is irrational. Is the multiset\\[\\{ \\lfloor \\alpha\\rfloor,\\lfloor 2\\alpha\\rfloor,\\lfloor 4\\alpha\\rfloor,\\ldots\\}\\cup \\{ \\lfloor \\beta\\rfloor,\\lfloor 2\\beta\\rfloor,\\lfloor 4\\beta\\rfloor,\\ldots\\}\\]complete? That is, can all sufficiently large natural numbers $n$ be written as\\[n=\\sum_{s\\in S}\\lfloor 2^s\\alpha\\rfloor+\\sum_{t\\in T}\\lfloor 2^t\\beta\\rfloor\\]for some finite $S,T\\subset \\mathbb{N}$?What if $2$ is replaced by some $\\gamma\\in(1,2)$?","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)"}