Quantitative weak mixing of self-affine tilings

We study the regularity of spectral measures of dynamical systems arising from a translation action on tilings of substitutive nature. The results are inspired in the work of A. Bufetov and B. Solomyak, where they previously established a log-H\"older modulus of continuity of one-dimensional self-similar tiling systems. We generalize this result to higher dimension in the more general case of self-affine tilings systems. Futher analysis, leads to uniform estimates in the whole space of spectral parameters, allowing to deduce logarithmic rates of weak mixing.