Abstract
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.
Highlights
Approximate subgroups were defined by Terence Tao [21] in order to give a noncommutative generalisation of results from additive combinatorics
We show a structure theorem for all approximate subgroups in soluble algebraic groups (Theorem 1)
Infinite approximate subgroups of soluble Lie groups which is commensurable to such a Λ is called a Meyer subset of G
Summary
Approximate subgroups were defined by Terence Tao [21] in order to give a noncommutative generalisation of results from additive combinatorics. Infinite approximate subgroups of soluble Lie groups which is commensurable to such a Λ is called a Meyer subset of G This construction was first introduced by Yves Meyer in the abelian case [15] and extended by Michael Björklund and Tobias Hartnick [2]. Motivated by this theorem the authors of [2] asked whether similar results would hold for other classes of locally compact groups [2, Problem 1.]. We answer this question in the soluble Lie case. Λ is a Meyer subset according to Theorem 1
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