Continued fractions with large prime partial quotients

The paper proves a zero–one law for the Lebesgue measure of E′_ℓ(φ) by (i) rewriting E′_ℓ(φ) as a limsup set under the Gauss map, (ii) expressing m(B_n) as a sum over level-ℓ cylinders with digits restricted to primes, m(I_ℓ(p_1,…,p_ℓ)) ≍ 1/(p_1^2⋯p_ℓ^2), and (iii) invoking a dynamical Borel–Cantelli lemma; the crucial analytic input is a new asymptotic for the tail of the almost-prime zeta function that yields m(B_n) ≍ (log log φ(n))^{ℓ−1}/(φ(n) log φ(n)) and hence the zero–one dichotomy via ∑ m(B_n) (Lemma 4.1 and Theorem 1.7 → Theorem 1.8). All these steps are laid out explicitly in the paper’s Section 4 and auxiliary results in Section 3 . The candidate solution establishes the same criterion with essentially the same structure—Gauss cylinders + Borel–Cantelli—while replacing the paper’s tail estimate (Theorem 1.7) by a Landau k-almost-prime count plus partial summation to obtain S_ℓ(y) ≍ (log log y)^{ℓ−1}/(y log y). This yields the same convergence/divergence test and, via a dynamical Borel–Cantelli lemma, the same zero–one law. Minor omissions in the model (e.g., explicitly handling the case when φ(n) fails to be eventually large) are handled carefully in the paper by a max-threshold device ψ(n)=max{T,φ(n)} (see the end of the proof of Theorem 1.8) . Net: both are correct; the proofs differ mainly in how the prime-sum tail is estimated.