LOCAL SPECTRAL ESTIMATES AND QUANTITATIVE WEAK MIXING FOR SUBSTITUTION Z-ACTIONS

The uploaded paper proves two main statements: (i) a dichotomy for primitive aperiodic substitutions with irreducible incidence matrix and no eigenvalues on the unit circle—either a non-trivial discrete spectral component exists or all mean-zero cylindrical functions have uniformly log-Hölder spectral measures (Theorem 2.2)—and (ii) for Salem-type substitutions, at algebraic spectral parameters ω ∈ Q(α), spectral measures satisfy pointwise Hölder bounds with constants depending as stated (Theorem 2.5) . The paper’s proof scheme hinges on (a) a standard inequality linking local spectral mass to L2-norms of twisted Birkhoff sums (Lemma 2.10) , (b) product/renormalization bounds for twisted sums via return words and the matrix/Riesz-product formalism (Proposition 4.1, with distances along (S^T)^n(ω·1) to a dual lattice) , and (c) a quantitative “Veech-type” orbit–lattice separation mechanism developed in Section 3, driving the log-decay of twisted sums in the no-unit-circle case and the power-law in the Salem case (see the “scheme of approximation” and its role in eq. (4.2), Lemma 3.2) . The candidate solution outlines exactly this strategy and arrives at the same conclusions, citing Host’s eigenvalue criterion to handle discrete spectrum (Theorem 2.1) and invoking the same renormalization/Product machinery and quantitative orbit separation. Minor gaps: the model glosses over the “gluing” near rational points q^{-1}Z and the special handling of ω=0 that the paper treats explicitly (Proposition 4.3 and (4.11)–(4.13)) before deducing the uniform version of (2.2) , and it informally posits a positive-frequency separation at a power threshold n^{−η} (stronger than what is directly stated), though it ultimately concludes only the log-Hölder bound consistent with Theorem 2.2. For the Salem case, the model’s description matches Theorem 2.5 and Remark 2.6, including the dependence of the Hölder exponent on |ω|, |σ0(ω)|, and the denominator L, with the leading constant depending only on the substitution and non-uniformity of the exponent across algebraic ω as later formalized by Theorem 2.7 .