Analytic linearization of a generalization of the semi-standard map: radius of convergence and Brjuno sum

The paper proves sharp lower and upper bounds for the linearization radius ρ of the C^2 skew-product (a generalization of the semi-standard map) in terms of the Brjuno sum of dα. Specifically, Theorem 1 shows ρ ≥ exp(−(2/d) B(dα) − C) without any phase assumption, and Theorem 2 shows ρ ≤ exp(−(2/d) B(dα) + C′) under the phase alignment Arg(ak) = kθ + π/2, with constants independent of α and monotone in the coefficients as stated ; the functional equation and coefficient recursions on which the proof is based are derived in Section 3 (notably equation (6)) and further used to isolate the small divisors Dl = −4 sin^2(π l α) . The upper bound relies on a carefully chosen subsequence of fast growing convergents for dα and a recursive function F that captures the contribution of the small divisors (Sections 4–6) . The candidate solution matches the statements but its proof hinges on a boundedness claim for the cohomological inverse Δζ−1 on a fixed Bruno-weighted coefficient norm that omits the necessary loss of analyticity; in this setting Δζ−1 is not bounded on a single-width space and one needs a width gap tied to 2B(dα). It also elides the semigroup-of-indices structure crucial in the paper’s argument and asserts an overly strong product estimate for small divisors. Consequently, while the end results coincide with the paper’s theorems, the model’s proof is incomplete/incorrect on key analytic estimates.