Stability of the Non-Critical Spectral Properties I: Arithmetic Absolute Continuity of the Integrated Density of States

The uploaded paper proves exactly the target statement: for v real-analytic, |λ| ≠ 1, there exists ε0(λ, v) > 0, independent of α, such that for all strongly Diophantine α and |ε| < ε0, the IDS of H^ε is absolutely continuous (Theorem 1.1; ε0 independent of α is emphasized in Remark 1.4) . The rotation number/IDS relation is explicitly N(E) = 1 − 2ρ(E) (equation (2.1)) . The supercritical proof uses homogeneity of the spectrum and a Herglotz-function criterion of Sodin–Yuditskii together with analyticity of the Lyapunov exponent and the identity ∂E L(E + iε) = Re G(0,0,E + iε) (Theorems 3.2–3.5) , while the subcritical case is handled by almost reducibility, yielding absolute continuity even for all irrational α (Theorem 4.1) . The candidate solution reaches the same conclusion and correctly cites this paper’s main theorem, but its Step 5 mischaracterizes the paper’s proof technique (periodic approximants and band-density L1 bounds); the paper’s approach instead hinges on homogeneity plus Lyapunov-exponent analyticity and Green’s-function analysis. The rotation-number normalization in the candidate ("N = ρ/π up to constant") is imprecise compared to N = 1 − 2ρ, but this does not affect the absolute-continuity conclusion. Overall, the paper’s result is correct and complete, and the model’s conclusion matches it, though via a different (and partially misattributed) proof sketch.