Regularity Properties of k-Brjuno and Wilton Functions

The PDF proves three statements: (i) for all k∈N and α∈[1/2,1], Bk,α∈BMO (Proposition 2.1), (ii) W=W1 is not in BMO (Theorem 2.2), and (iii) for α∈[1/2,g] with g=(√5−1)/2, Wα∈BMO (Theorem 2.3). The proof strategy is: invertibility of 1−Tk,1/2 on a periodic-even BMO-type space X∗ with spectral radius bound (√2−1)k (Proposition 2.4), transfer to general α via bounded-difference estimates (Propositions 2.5 and 2.8), a direct oscillation lower bound ~log n for W on symmetric intervals (using a standard identity for consecutive intervals), and a comparison W1/2−Wα∈L∞ for α≤g (Proposition 2.9), which with invertibility of 1−S(1/2) yields BMO for Wα. All of these are stated and proved in the paper (Proposition 2.1, Theorems 2.2–2.3, with supporting Propositions 2.4–2.9) . The candidate solution reaches the same conclusions but argues via an operator S_{k,α}f=x^k(f∘Aα), claiming spectral radius <1 on a similar X∗ for all α∈[1/2,1], then applying a Neumann series directly for each α, and treating Wilton similarly with (I+S_{1,α})Wα=L. This is a different (but standard) functional-analytic route; the paper itself only establishes spectral bounds and invertibility explicitly for α=1/2 and transfers to general α by bounded comparison, not by proving spectral radius <1 for each α. The model’s spectral-radius-for-all-α claim is not used in the paper and would need an external reference (it cites LMNN 2007), but it does not contradict the paper’s results. Hence, both are correct, with different proofs.