A Halmos–von Neumann theorem for model sets, and almost automorphic dynamical systems

Summary

Abstract. A subset Σ ⊆ ℝ^d is the spectrum of a model set Λ ⊆ ℝ^d if and only if it is a countable subgroup. The same result holds for Σ ⊆ Ĝ for a large class of locally compact abelian groups G.

Introduction

A model set Λ is a special kind of uniformly discrete and relatively dense subset of a locally compact abelian group G. Model sets were first studied systematically in 1972 by Yves Meyer, who considered them in the context of Diophantine problems in harmonic analysis. More recently, model sets have played a prominent role in the theory of quasicrystals, beginning with N. G. de Bruijn's 1981 discovery that the vertices of a Penrose tiling are a model set. Much of the interest in model sets is due to the fact that although they are aperiodic, model sets have enough “almost periodicity” to give them a discrete Fourier transform. This corresponds to spots, or Bragg peaks, in the X-ray diffraction pattern of a quasicrystal.

In this paper, we study model sets from the point of view of ergodic theory and topological dynamics. A translation invariant collection X of model sets in G has a natural topology, and with respect to this topology the translation action of G on X is continuous. We think of this action as a model set dynamical system.

Model set dynamical systems are closely related to tiling dynamical systems (see, and the first examples that were worked out, Penrose tilings, “generalized Penrose tilings”, and chair tilings, were tilings with model sets as vertices.