Infinite approximate subgroups of soluble Lie groups

We study infinite approximate subgroups of soluble Lie groups. We show that approximate subgroups are close, in a sense to be defined, to genuine connected subgroups. Building upon this result we prove a structure theorem for approximate lattices in soluble Lie groups. This extends to soluble Lie groups a theorem about quasi-crystals due to Yves Meyer.

Automated summary

The study investigates infinite approximate subgroups of soluble Lie groups, showing their proximity to genuine connected subgroups in a defined sense. Building on this, a structure theorem for approximate lattices in soluble Lie groups is proved, extending Yves Meyer's theorem on quasi-crystals to the context of soluble Lie groups.