An approximate subgroup $X$ of a 2nd countable locally compact group $G$ is called an approximate lattice if it is closed and discrete, and has finite volume in $G$, with $XB=G$. This slightly generalizes two definitions due to Michael Björklund and Tobias Hartnick, and studied by Simon Machado. In the commutative case, they were classified by Yves Meyer in 1972, in what late became the mathematical theory of quasicrystals.

Let $G$ be a semisimple algebraic group, such as $SL_2$. Arithmetic lattices are classically constructed from the group $G(\mathbb{Q})$ of rational points of $G$ by intersecting with compact open subgroup $W$ of the identity in cofinitely many places and projecting to the product of the remaining places; for instance one may obtain $SL_2(\mathbb{Z})$ within $SL_2(\mathbb{R})$ with $W=\prod_p SL_2(\mathbb{Z}_p)$. Björklund and Hartnick showed that the same cut-and-project construction, but with an arbitrary neighborhood of $1$ allowed as $W$, leads to approximate lattices. Employing the general structure theorem for approximate subgroups along with Margulis' arithmeticity theorems, we show that any discrete approximate subgroups is commensurably contained either in a discrete group, a proper Zariski closed subgroup, or in an approximate lattice of such arithmetic origin . Essential arithmeticity for approximate lattices continues to hold in wide family of locally compact groups.