Explore the intricate world of Pisot substitutions and their impact on self-similar inflation tilings in Euclidean space through this 48-minute lecture by Michael Baake. Delve into the complex relationship between dynamical systems' spectra and Fourier-Bohr coefficients, focusing on the challenges of computing and analyzing Rauzy fractals' Fourier transforms. Discover a novel cocycle approach to a matrix Riesz product formula, enabling efficient and precise computations of these transforms. Examine the connection between uniform distribution results and the Eberlein decomposition of autocorrelation measures, leading to explicit spectral decompositions. Cover topics such as diffraction theory, pure point spectra, complex windows, and various examples including the Silver mean diffraction and Ammann-Beenker patterns.