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Given a cut and project scheme and a pre-compact Borel window we show thatalmost surely all positions of the window give rise to point sets withBesicovitch almost periodic Dirac combs. In particular, all those positionslead to pure point diffractive point sets with autocorrelation given by thecovariogram of the window and diffraction given by the square of the Fouriertransform of the window. Moreover the Fourier--Bohr coefficients exist and theintensity of the Bragg peaks is given by their absolute value square. We showthe existence of an ergodic measure with pure point dynamical spectrum suchthat all those window positions give rise to point sets which are generic forthis measure. We complete the paper by reanalysing the class of weak model setsof maximal density.

Model sets with precompact Borel windows