Metric Theory for Continued Fractions with Multiple Large Partial Quotients

The paper states and proves a 0–1 law for Lebesgue measure and a three‑phase Hausdorff dimension formula for F(r,ψ) that depend on the monotone envelope ψ̃(n)=min_{m≥n}ψ(m) and the growth parameters B and b (Theorems 1.7 and 1.8) . The candidate solution derives the same statements: (i) a measure criterion equivalent to ∑ n^{r−1}ψ̃(n)^{-r}, and (ii) a pressure formula dim_H F(r,ψ) = inf{s ≥ 0 : P(T, −s log|T′| − (s+(2s−1)(r−1)) log B) ≤ 0} for 1 < B < ∞, with full dimension at B=1 and 1/(1+b) at B=∞. The paper’s proof of the divergence direction for measure uses a careful dyadic/separation construction (Um, Fm) and Petrov’s lemma to handle quasi‑independence (Proposition 2.2 for mixing; Proposition 2.3; Lemmas 3.2–3.3) , while the model appeals directly to Kochen–Stone with an unproven global quasi‑independence bound. For dimension, the paper’s upper bound is obtained via an explicit cylinder‑cover computation leading to the same pressure balance (formulas (4.6)–(4.7)) and the lower bound is supplied by a result of Hussain–Shulga through sets Gr(α,A) (Theorem 4.2) and, for B=∞, by the Luczak/Feng–Wu–Liang–Tseng dimension 1/(1+b) (Theorem 4.3) . Thus, the statements match and the strategies are conceptually similar (pressure + mixing/Borel–Cantelli), but the model omits several technical steps that the paper treats carefully. Overall: both correct, with different levels of rigor and different technical tools.