Model sets with precompact Borel windows

Given a cut and project scheme and a precompact Borel window, we show that almost surely all positions of the window give rise to point sets with Besicovitch almost periodic Dirac combs. In particular, all those positions lead to pure point diffractive point sets with autocorrelation given by the covariogram of the window and diffraction given by the inverse Fourier transform of the window squared. Moreover, the Fourier–Bohr coefficients exist and the intensity of the Bragg peaks is given by their absolute value squared. We show the existence of an ergodic measure with pure point dynamical spectrum such that all those window positions give rise to point sets which are generic for this measure.