Proximality and pure point spectrum for tiling dynamical systems

Abstract

We investigate the role of the proximality relation for tiling dynamical systems. Under two hypotheses, namely that the minimal rank is finite and the set of fiber distal points has full measure we show that the following conditions are equivalent: (i) proximality is topologically closed, (ii) the minimal rank is one, (iii) the continuous eigenfunctions of the translation action span the L^2-functions over the tiling space. We apply our findings to model sets and to Meyer substitution tilings. It turns out that the Meyer property is crucial for our analysis as it allows us to replace proximality by the a priori stronger notion of strong proximality.