  Approximate lattices are aperiodic generalisations of lattices of locally
compact groups that were first studied in seminal work of Yves Meyer. They are
defined as those uniformly discrete approximate subgroups (symmetric subsets
stable under multiplication up to a finite error) of locally compact groups
that have finite co-volume. Meyer showed that approximate lattices of Euclidean
spaces (a.k.a. Meyer sets) are related to lattices in higher-dimensional
Euclidean spaces via the cut-and-project construction. A fundamental challenge
of the theory of approximate lattices is to extend Meyer's theorem beyond
Euclidean spaces.
  Our main result provides a complete structure theorem for approximate
lattices valid in all linear algebraic groups over local fields and their
finite products, in particular providing the most general extension of Meyer's
theorem to date. Our proof relies on an extension of a theorem of
Lubotzky--Mozes--Raghunathan to approximate lattices in S-adic semi-simple
groups, a notion of cohomology tailored to the study of approximate subgroups,
a universality statement complementing a recent result of Hrushovski and a
generalisation of a result of Burger and Monod about bounded cohomology of
lattices.
