Complexity of Cut-and-Project Sets of Polytopal Type in Special Homogeneous Lie Groups

TL;DR

This paper investigates the complexity growth of cut-and-project sets in non-abelian homogeneous Lie groups, establishing asymptotic behavior and generalizing acceptance domains to broader group classes.

Contribution

It provides the first asymptotic complexity results for polytopal cut-and-project sets in non-abelian homogeneous Lie groups and extends the concept of acceptance domains.

Findings

Complexity function grows like r^{homdim(G) * dim(H)}.

Results apply to two-step nilpotent Lie groups.

Generalization of acceptance domains to locally compact second countable groups.

Abstract

The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups we can establish that the complexity function asymptotically behaves like $r^{homdim(G) dim(H)}$. Further we generalize the concept of acceptance domains to locally compact second countable groups.