Small doubling for discrete subsets of non-commutative groups and a theorem of Lagarias

Abstract

Approximate lattices of Euclidean spaces, also known as Meyer sets, are aperiodic subsets with fascinating properties. In general, approximate lattices are defined as approximate subgroups of locally compact groups that are discrete and have finite co-volume. A theorem of Lagarias provides a criterion for discrete subsets of Euclidean spaces to be approximate lattices. It asserts that if a subset $X$ of $\mathbb{R}^n$ is relatively dense and $X - X$ is uniformly discrete, then $X$ is an approximate lattice. We prove two generalisation of Lagarias' theorem: when the ambient group is amenable and when it is a higher-rank simple algebraic group over a characteristic $0$ local field. This is a natural counterpart to the recent structure results for approximate lattices in non-commutative locally compact groups. We also provide a reformulation in dynamical terms pertaining to return times of cross-sections. Our method relies on counting arguments involving the so-called periodization maps, ergodic theorems and a method of Tao regarding small doubling for finite subsets. In the case of simple algebraic groups over local fields, we moreover make use of deep superrigidity results due to Margulis and to Zimmer.