On the structure of the dimension spectrum for continued fraction expansions

The paper proves the full-dimension-spectrum result (Theorem 1.1) under the submultiplicativity hypothesis a_{n+m} ≤ a_n a_m via Perron–Frobenius operators and a strict-break-point criterion, and the argument is complete and consistent with the stated assumptions . The candidate solution attempts a different, elementary proof (covering sums U_B, L_B and a greedy sub-sum construction), but it contains critical gaps: (i) it incorrectly asserts the existence of finite subsets F_k with min(F_k)→∞ whose U_{B_k}(s)→1 even in cases where ∑ a_n^{-2s} converges (e.g., A = {q^n}); this is impossible because tails then carry vanishing mass; and (ii) it inverts the monotonicity when bracketing the dimension by the U/L-criteria (claiming t_k^+ > s when U_{B_k}(s) < 1 actually implies t_k^+ < s). These errors obstruct the proof as written.