Model sets with Euclidean internal space

TL;DR

This paper provides a new dynamical characterization of inter-model sets with Euclidean internal space, extending previous results by incorporating additional algebraic and flow conditions related to the internal differences and torus dynamics.

Contribution

It introduces two new conditions involving the rank of the internal difference group and a flow on a torus, offering an almost dynamical characterization similar to prior work.

Findings

Characterization of inter-model sets with Euclidean internal space.

Inclusion of rank conditions on internal difference groups.

Use of torus flows to describe maximal equicontinuous factors.

Abstract

We give an almost dynamical characterization of inter-model sets with Euclidean internal space. This characterization is similar to previous results for general inter-model sets obtained independently by Baake, Lenz and Moody, and Aujogue. The new ingredients are two additional conditions on the rank of the Abelian group generated by the set of internal difference and a flow on a torus defined via the address map introduced by Lagarias that play the role of the maximal equicontinuous factor in the previous characterizations.