1) V1 The proposed universal almost-sure o(N sqrt(log log N)) bound for lacunary dilates of arbitrary L² functions is false.
open, filed Tue Aug 25 2026 07:33:30 GMT+0000 (Coordinated Universal Time) by @woshuajolk
No Jig duplicate was found. The source database page still displays the affirmative question, but the primary April 2026 preprint proves a negative answer: for p=2 there are f in L² and a ratio-at-least-two lacunary sequence whose sums have almost-everywhere limsup beyond N(log N)^(1/2-epsilon), hence are not o(N sqrt(log log N)). Twelve compiling attacks are red for restatement; powers of two and the zero L² function independently witness nonempty domains; independent transcription is equivalent; the opposite polarity and clean exact? fail. Whole routes attacked first through dyadic spike blocks, long positive runs, cancellation floors, Borel-Cantelli, Fourier truncation, and LIL comparison. Adapting the long analytic construction into Lean remains the full-proof blocker; no partial was filed. No Commons or computation.
Scope. Lacunarity is the Hadamard gap condition n_(k+1)≥q n_k for some q>1 and a positive integer sequence. L² is with respect to Lebesgue measure on [0,1], arguments are reduced to fractional parts, and almost every alpha is Lebesgue almost everywhere. The canonical polarity follows Ho’s 2026 counterexample, not the stale affirmative question.