SYSTEMS OF RANK ONE, EXPLICIT ROKHLIN TOWERS, AND COVERING NUMBERS

Weiß proves for irrational rotations that (i) if the continued-fraction tail is eventually all ones, then F2(fα) = (4ϕ+3)/10, and (ii) if the tail is eventually >1, then F2(fα) = F1(fα). These are stated as Theorem 1.6 and derived via a detailed two-interval endpoint-orbit analysis based on a refined Three-Gap theorem and Lemma 2.2, which gives the needed piecewise-linear constraints and matching constructions. The model’s solution reproduces exactly these two cases with the same limiting value (4ϕ+3)/10 in the ϕ-tail case and the collapse F2=F1 otherwise, and it invokes the same structural ingredients (Three-Gap theorem, continued fractions, and an endpoint-orbit/tower packing argument). For F1, the paper recalls Chekhova’s formula F1(fα) = lim q_{n+1}||q_n α|| (equivalently limsup 1/(1+t_nv_n)), consistent with the model’s envelope formulation limsup max{q_k||q_{k-1}α||, q_{k+1}||q_kα||}. Hence the statements and the proof skeletons are in agreement. See Theorem 1.5 for F1, Lemma 2.2 for the two-interval constraints, and Theorem 1.6 for the exact values of F2 in both regimes in the paper 2106.10054.pdf .