Quantitative weak mixing for random substitution tilings

For $N$ compatible substitution rules on $M$ prototiles $t_1,\dots,t_M$, consider tilings and tiling spaces constructed by applying the different substitution rules at random. These give (globally) random substitution tilings. In this paper I obtain bounds for the growth on twisted ergodic integrals for the $\mathbb{R}^d$ action on the tiling space which give lower bounds on the lower local dimension of spectral measures. For functions with some extra regularity, uniform bounds on the lower local dimension are obtained. The results here extends results of Bufetov-Solomyak to tilings of higher dimensions.