Lochs-type theorems beyond positive entropy

The paper’s Theorem 6.8 proves that for two a.e. log-balanced sequences of partitions P1,P2 with weights f1,f2 satisfying (i) sum_{n} e^{-δ f1(n)} < ∞ for all δ>0, (ii) f2 nondecreasing, and (iii) f2(n+1)−f2(n)=o(f2(n)), one has f2(Ln(x,P1,P2))/f1(n) → 1 a.e. The proof splits into limsup ≤ 1 (Proposition 6.6) and liminf ≥ 1 (Proposition 6.7), with a key counting bound λ(Dn,ε) ≤ 2 exp(−(1−ε) f1(n)+(1+ε) f2(m(n))) and a Borel–Cantelli step; the summability condition (31) is verified using (iii) (Lemma 6.5), completing the proof of Theorem 6.8 . The candidate solution reproduces the same strategy: an Egorov-uniformization supplying two-sided exponential bounds for interval measures, a standard counting/endpoint argument yielding the same exponential measure estimate for the ‘bad’ set, and Borel–Cantelli to force Ln(x) above a carefully chosen m(n). It then uses monotonicity of f2 and the small-increment hypothesis (iii) to control the discretization gap and conclude liminf ≥ 1, while the limsup follows directly from inclusion and the uniform bounds—exactly as in Proposition 6.6 . Minor differences are present only in packaging (Egorov blockwise vs. ε-good intervals and the abstract summability condition), but the core combinatorial and probabilistic steps are the same. One minor slip in the model is an unnecessary appeal to “P2 self-refining” and to the monotonicity of Ln in n; Theorem 6.8 does not assume self-refinement and such monotonicity of Ln is not needed, with finiteness of Ln and the a.e. limit ensured by the log-balanced hypotheses and the counting/Borel–Cantelli argument .