{"schema":"vela.problem-packet.v0.1","problem":996,"statement":"Let $n_1<n_2<\\cdots$ be a lacunary sequence of integers, and let $f\\in L^2([0,1])$. Let $f_n$ be the $n$th partial sum of the Fourier series of $f(x)$. Is there an absolute constant $C>0$ such that, if\\[\\| f-f_n\\|_2 \\ll \\frac{1}{(\\log\\log\\log n)^{C}}\\]then\\[\\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{k\\leq N}f(\\{\\alpha n_k\\})=\\int_0^1 f(x)\\mathrm{d}x\\]for almost every $\\alpha$?","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)"}