We study generalisations of a theorem of Yves Meyer concerning the structure of approximate lattices. An approximate lattice is a discrete approximate subgroup of a locally compact group - i.e. a subset that is closed under multiplication up to an error controlled by a finite set - that has finite co-volume. We study, successively, generalisations of Meyer's theorem in soluble Lie groups, in amenable locally compact groups and in higher-rank semi-simple algebraic groups.

Along the way, we investigate properties of closed and discrete approximate subgroups of locally compact groups in general.