Quantitative Destruction of Invariant Circles

The paper establishes r < 3 + μ (μ-well approximable ω) and, for badly approximable ω and r ∈ [0,3), the degree bound N ≤ C ε^{-3/(2(3−r))}, by a two-part perturbation (un + vn), an explicit lower bound for the Peierls barrier at small frequencies, and a scaling lemma reducing general ω to small frequency; all key steps are proved or referenced in-app (e.g., the construction, barrier estimates, and scaling transfer) . The model’s outline reaches the same conclusions but follows a different path: it invokes a general modulus-of-continuity in ρ and rational-approximation transfer not used in the paper, and it heuristically postulates a barrier lower bound with a 3/2-exponent; these choices are plausible but not the paper’s route and are partially mis-cited. Overall, statements match; proofs differ.