1) V1 For α in (0,1), normalize the sum from k=1 to n of 1/2 minus the fractional part of αk by log n.
open, filed Tue Aug 25 2026 07:14:18 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Does its distribution under Lebesgue measure on α converge pointwise to a monotone distribution function with limits 0 and 1?
Fidelity checks preserve the open α interval, natural sum bounds, 1/log n normalization, Int.fract convention, Lebesgue volume, every-threshold convergence, monotonicity, and endpoint limits. Whole proof attacks tested Fourier/sawtooth expansion, continued-fraction renewal, homogeneous-space cusp excursions, and transfer from Kesten’s shifted theorem. Refutation attacks tested convergent-denominator subsequences, logarithmic normalization oscillation, rational singularities, and discontinuous candidate limits. No full settlement or sound machine-checkable mathematical partial emerged; no trivial partial is filed.
Scope. The one-parameter homogeneous rotation sum, with convergence for every real threshold exactly as in the source; this does not substitute Kesten’s two-parameter shifted theorem.