Invariant Densities for Random Continued Fractions

The paper states in Theorem 1.2 that the essential spectral radius on C^k([0,1]) satisfies ress(L_p|_{C^k}) ≤ ζ(2k+2)^{−min(p,1−p)}) and that L_p is bounded on C^k, aligning with the solver’s target claim. However, the argument actually worked out in Section 3.1 yields the different upper bound ress(L_p|_{C^k}) ≤ ζ(2k+2) − min(p,1−p), from which quasi-compactness is only deduced for sufficiently large k, not for all k; the stronger exponent-type bound is asserted but not proved in the provided text (compare Theorem 1.2 with Section 3.1’s calculation) . The model’s alternative proof sketches a Lasota–Yorke/Doeblin–Fortet route, but it contains substantive gaps: an unjustified Doeblin-type minorization of the normalized operator W_k by a rank-one ‘atom’, a scaling slip in passing from U_k to W_k that leads to an exponent mismatch, and an incorrect inequality direction in the concavity step; consequently the claimed bound ζ(2k+2)^{−min(p,1−p)} is not established by the model. In short, the paper’s detailed proof supports the weaker ζ(2k+2) − min(p,1−p) bound and quasi-compactness for large k (as they explicitly state), while the stronger exponent bound is unproven; the model’s argument also does not currently supply a correct proof of the stronger claim. Therefore, both are incomplete relative to the stated stronger result. See the paper’s statement of Theorem 1.2 (exponent bound) and its Section 3.1 derivation (additive bound) for this discrepancy, as well as their general Theorem 3.1 and quasi-compactness framework via Nussbaum’s formula and positivity arguments .