{"schema":"vela.problem-packet.v0.1","problem":999,"statement":"For any function $f:\\mathbb{N}\\to \\mathbb{N}$ the property that, for almost all $\\alpha$\\[\\left\\lvert \\alpha-\\frac{p}{q}\\right\\rvert < \\frac{f(q)}{q}\\]has infinitely many solutions with $(p,q)=1$, is equivalent to\\[\\sum_{q\\geq 1}\\phi(q)\\frac{f(q)}{q}=\\infty.\\]","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)"}