Fourier Transform and Expanding Maps on Cantor Sets

The paper’s main theorem explicitly requires a dimension threshold: if μ is an equilibrium state for a potential with exponentially decaying variations and dim_H μ is close enough to dim_H K, then μ̂(ξ) decays at a polynomial rate (Theorem 1.1; see the statement and discussion around it) . This dependence on the dimension is underscored in Remark 1.2(2), which explains the need for dim_H μ to be sufficiently close to s0 = dim_H K in order to apply Naud’s C^1 contraction for complex transfer operators . The candidate solution, however, asserts the result for all equilibrium (Gibbs) measures under (1)–(4), and even claims it holds “a fortiori” for any ε0>0; this contradicts the paper’s stated assumptions and proof strategy. Moreover, the candidate attributes to Sahlsten–Stevens a uniform oscillatory transfer-operator sup bound sup_y |L_ϕ^n(e^{-2πiξ·})(y)| ≤ C|ξ|^{−α} for n≈c log|ξ|, which is not what the paper proves. The paper instead derives Fourier decay for μ̂(ξ) via a combination of large deviations for Gibbs measures, non-concentration of distortions under total non-linearity, sum–product estimates, and spectral bounds for the complex Ruelle operator L_{−sτ} at s=δ−2πiξ, with δ near s0 (see e.g. the use of Naud’s contraction and identity (4.6)) , and completes the proof with the final decay estimate |μ̂(ξ)| = O(|ξ|^{−α}) for some α>0 . The paper’s assumptions (1)–(4) are stated clearly (uniform expansion, Markov property, bounded distortion, total non-linearity) , and the role of total non-linearity in ensuring non-concentration is emphasized . In short, the paper is consistent and careful about the dimension requirement; the model overclaims beyond what the paper establishes.