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 the theory of discrete subgroups of Lie groups, ergodic theory and additive combinatorics.

 

A central question arising from seminal work of Yves Meyer asks whether approximate lattices have an arithmetic origin. I will present a complete structure theorem for approximate lattices in linear algebraic groups in terms of bounded cohomology that, in particular, answers this question. I will pinpoint key instances where the interplay between additive combinatorics and other fields - e.g. ergodic theory and algebraic groups - is particularly fruitful.