Approximate Lattices in Algebraic Groups

Approximate lattices in locally compact groups are approximate subgroups that are discrete and have finite co-volume. They provide natural examples of objects at the intersection of algebraic groups, ergodic theory and additive combinatorics... with a twist. Indeed, approximate lattices are note finite - rendering most of the known theory of approximate subgroups unavailable to us.

A central question arising from seminal work of Yves Meyer asks whether all approximate lattices have an arithmetic origin. After presenting a complete structure theorem for approximate lattices in algebraic groups generalising arithmeticity theorems of Margulis and Mostow, I will pinpoint key instances where the interplay between additive combinatorics and other fields - e.g. ergodic theory and algebraic groups - is particularly fruitful.