{"schema":"vela.problem-packet.v0.1","problem":232,"statement":"For $A\\subset \\mathbb{R}^2$ we define the upper density as\\[\\overline{\\delta}(A)=\\limsup_{R\\to \\infty}\\frac{\\lambda(A \\cap B_R)}{\\lambda(B_R)},\\]where $\\lambda$ is the Lebesgue measure and $B_R$ is the ball of radius $R$.Estimate\\[m_1=\\sup \\overline{\\delta}(A),\\]where $A$ ranges over all measurable subsets of $\\mathbb{R}^2$ without two points distance $1$ apart. In particular, is $m_1\\leq 1/4$?","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)"}