On upper and lower fast Khintchine spectra of continued fractions

The paper proves Theorem 1.2: dim_H Ē(ψ) = 1/(b+1) and dim_H E_(ψ) = 1/(B+1), where log b = lim inf (log ψ(n))/n and log B = lim sup (log ψ(n))/n. It establishes non-emptiness, the upper bound for Ē via a limsup set F(ψ) and Lemma 2.7, including the delicate b = 1 case handled through the set Π_∞ with dim_H = 1/2, and the lower bound via a carefully constructed nondecreasing minorant and a product-based construction using Lemma 2.4; an analogous construction yields the result for E_(ψ) through sequences T_j and c_n with (2.12)–(2.14) (Theorem 1.2 and its proof outline: ; upper-bound strategy and Lemma 2.7: ; the b=1 case via Lemma 2.3: ; use of Lemma 2.4 and product method: ; lower-bound sequence construction for Ē: ; liminf-case construction and bounds (2.12)–(2.14): ). By contrast, the candidate’s solution relies on an unproved and generally false identification lim inf (Δ_{n+1}/Δ_n) = b (with Δ_n = ψ(n)−ψ(n−1)), and on a non-cited dimension formula for limsup digit-threshold sets attributed to Mauldin–Urbański; these are not justified for arbitrary ψ and fail exactly in the regime b = 1, which the paper covers. The lower-bound “block” construction in the model is also too sketchy to verify. Hence the paper’s argument is correct and complete, while the model’s outline contains critical gaps.