{"schema":"vela.problem-packet.v0.1","problem":1179,"statement":"Let $0<\\epsilon<1$ and let $g_\\epsilon(N)$ be the minimal $k$ such that if $G$ is an abelian group of size $N$ and $A\\subseteq G$ is a uniformly random subset of size $k$, and\\[F_A(g) = \\#\\left\\{ S\\subseteq A : g = \\sum_{x\\in S}x\\right\\},\\]then, with probability $\\to 1$ as $N\\to \\infty$,\\[\\left\\lvert F_A(g)-\\frac{2^k}{N}\\right\\rvert \\leq \\epsilon \\frac{2^k}{N}\\]for all $g\\in G$.Estimate $g_\\epsilon(N)$ - in particular, is it true that for all $\\epsilon>0$\\[g_\\epsilon(N)=(1+o_\\epsilon(1))\\log_2N?\\]","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)"}