Local spectral estimates and quantitative weak mixing for substitution $\mathbb{Z}$-actions

The paper investigates H\"older and log-H\"older regularity of spectral measures for weakly mixing substitutions and the related question of quantitative weak mixing. It is assumed that the substitution is primitive, aperiodic, and its substitution matrix is irreducible over the rationals. In the case when there are no eigenvalues of the substitution matrix on the unit circle, our main theorem says that a weakly mixing substitution $\mathbb{Z}$-action has uniformly log-H\"older regular spectral measures, and hence admits power-logarithmic bounds for the rate of weak mixing. In the more delicate Salem substitution case, our second main result says that H\"older regularity holds for algebraic spectral parameters, but the H\"older exponent cannot be chosen uniformly.