An approximate subgroup is a subset X of a group G, such that 1 ∈ X, X = X−1, and such that the set of products X · X = {x · y : x, y ∈ X} is ‘almost’ equal to X, more precisely it is contained in X · F for some finite F ⊂ G. Approximate subgroups arise in many areas of analysis, combinatorics and geometry, as well as in model theory. The finite ones were classified by Breuillard, Green and Tao; they essentially arise in nilpotent groups. An approximate lattice in G = Rn, or in the matrix group GLn(R), is a discrete approximate subgroup X that has finite covolume; i.e. there exists a subset D ⊂ G of finite measure, with XD = G. Approximate lattices in Rn were classified by Meyer in the 1970’s, and eventually became the mathematical model for quasicrystals. I will present a generalization to semisimple groups; in effect all irreducible approximate lattices have arithmetic origin. They arise from number fields via a classical construction of Borel-Harish-Chandra; the approximate setting allows greater flexibility in putting archimedean and non-archimedean places on the same footing. The proof uses a construction arising naturally from basic questions in model theory (amalgamation, the Lascar group).

院士讲坛34.pdf