QUANTITATIVE REDUCIBILITY OF Ck QUASI-PERIODIC COCYCLES

The paper’s main theorem (Theorem 1.1) states that for α ∈ DCd(κ,τ), k > 14τ + 2, and sufficiently small ∥f∥k, if the fibered rotation number ρ(α,Ae^f) is Diophantine or rational w.r.t. α, then the SL(2,R) cocycle (α,Ae^f) is C^{k,k0}-reducible with k0 < k − 10τ − 3, with the conjugacy defined on 2Td. This is stated explicitly and consistently in the introduction and preliminaries . The proof proceeds via an analytic KAM step with a resonant/non-resonant dichotomy and explicit estimates (Proposition 3.1), followed by an analytic-approximation/smoothing scheme and an arithmetic analysis of resonances using the fibered rotation number and degree relation ρ(α,A2) = ρ(α,A1) − ⟨deg B,α⟩/2 under conjugacy . In the rational case they track the degree through resonant steps to reach a non-resonant regime . The candidate solution mirrors this structure: it uses an analytic KAM step with a resonant/non-resonant alternative, smoothing of C^k data, arithmetic control via the rotation number (including period-doubling), and a controlled derivative loss to obtain exact C^{k,k0} reducibility. Differences are minor (e.g., the candidate uses a standard Fourier-truncation smoothing and cites classical sources, whereas the paper develops specific quantitative constants and a refined SU(1,1) setup), but the logical pathway and conclusions coincide.