Determining $\mathbb{R}$-Rank in Semisimple Lie Groups via uniform approximate Lattice arising as Regular Model Sets

Abstract

Let $G$ be a linear semisimple Lie group without compact factors. We show that uniform approximate lattices $\hat{I}›$ arising as regular model sets in $G$ determine the ambient group $G$ in a strong sense. Specifically, for every non-compact Cartan subgroup $C$ of $G$, there exists $g\in G$ such that the intersection $gC{g}^{-1}\cap \hat{I}{›}^2$ is non-empty and itself forms a uniform approximate lattice, extending a classical result of Mostow for lattices. The proof relies on a Moore-type ergodicity theorem for the hull of a strong approximate lattice, proved here as a key tool. Moreover, we prove that such approximate lattices determine the $\mathbb{R}$-rank of the ambient group $G$, drawing on ideas from the work of Prasad and Raghunathan on lattices.

18 pages