Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks

We study uniquely ergodic dynamical systems over locally compact, sigma-compact Abelian groups. We characterize uniform convergence in Wiener/Wintner type ergodic theorems in terms of continuity of the limit. Our results generalize and unify earlier results of Robinson and Assani respectively. We then turn to diffraction of quasicrystals and show how the Bragg peaks can be calculated via a Wiener/Wintner type result. Combining these results we prove a version of what is sometimes known as Bombieri/Taylor conjecture. Finally, we discuss various examples including deformed model sets, percolation models, random displacement models, and linearly repetitive systems.