Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

Abstract

ABSTRACT. We introduce and study model sets in commutative spaces, i.e. homogeneous spaces of the form G/K where G is a (typically non-abelian) locally compact group and K is a compact subgroup such that (G, K) is a Gelfand pair. Examples include model sets in hyperbolic spaces, Riemannian symmetric spaces, regular trees and generalized Heisenberg groups. Continuing our work from [6] we associate with every regular model set in G/K a Radon measure on K\G/K called its spherical auto-correlation. We then define the spherical diffraction of the regular model set as the spherical Fourier transform of its spherical auto-correlation in the sense of Gelfand pairs. The main result of this article ensures that the spherical diffraction of a uniform regular model set in a commutative space is pure point. In fact, we provide an explicit formula for the spherical diffraction of such a model set in terms of the automorphic spectrum of the underlying lattice and the underlying window. To describe the coefficients appearing in this formula, we introduce a new type of integral transform for functions on the internal space of the model set. This integral transform can be seen as a shadow of the spherical Fourier transform of physical space in internal space and is hence referred to as the shadow transform of the model set. To illustrate our results we work out explicitly several examples, including the case of model sets in the Heisenberg group.

Delone dynamical systems and spectral convergence

“…The following proposition is a well-known statement from abstract harmonic analysis. Our proof follows the lines of the proof of [22,Proposition 4.8]. PROPOSITION 4.5.…”

“…Thus, Del U,K ⊆ C (G) is a compact, second countable Hausdorff space endowed with the induced Chabauty-Fell topology, cf. [5,22,73,88]. Note that due to compactness, we have K (Del U,K ) = C (Del U,K ).…”

“…Concerning the abelian and in particular the Euclidean situation, we refer to [3] and references therein. A list of examples in non-abelian groups can be found in [22]. We extensively use a parametrization map introduced in the abelian case by [88] ('torus parametrization') and in the non-abelian world by [22].…”

“…For a detailed introduction and overview, we refer further to the recent monograph [3] and references therein. Recently, a new noncommutative spherical diffraction theory was carried out in [21,22] for regular model sets. These results will allow us to approximate the autocorrelation even in non-abelian settings.…”

“…This fact relies on the parametrization map relating the corresponding Delone dynamical system with the homogeneous type dynamics arising from the cut-and-project scheme. For abelian groups, this connection has been established by Schlottmann [88], and for general locally compact, second countable groups, this has recently been shown in [21,22]. To deal with the natural discontinuities of the cutting, we approximate the characteristic function χ W of the window by continuous functions (called window functions) with compact support in H .…”