Minimal distances in quasicrystals

A general expression is derived for the minimal distance epsilon(Omega) between points of a cut and project quasicrystal Sigma(Omega) in R-n with a convex acceptance window Omega. The study of minimal distances amounts to the study of one-dimensional quasicrystals and their rescalings which occur in Sigma(Omega). For an n-dimensional ball as Omega, the exact value of epsilon(Omega) is calculated for any radius; for Omega 'close' to a ball, a simple formula is given; for all Omega upper and lower bounds for epsilon(Omega) are found. The latter are easy to use even when Omega is of a complicated shape.