Combinatorics Seminar - Simon Machado

16:15-18:00

Speaker: Simon Machado

Title: When are discrete subsets of Lie groups approximate subgroups ? Around a theorem of Lagarias.

Abstract: 

Meyer sets are fascinating objects: they are aperiodic subsets of Euclidean spaces that nonetheless exhibit long-range aperiodic order. Sets of vertices of the Penrose tiling (P3) and Pisot—Vijarayaghavan numbers of a real number field are some of the most well-known examples. In modern lingo, they can be defined as the discrete and co-compact approximate subgroups of Euclidean spaces.

Lagarias found an elegant characterisation of Meyer sets: they are those subsets X of a Euclidean space E that are relatively dense (any point of E is within uniformly bounded distance of X) and whose Minkowski difference X - X is uniformly discrete (the distance between any two points is uniformly bounded below). In essence, his theorem provides a characterisation of discrete approximate subgroups that is analogous to the Plunnecke—Ruzsa theorem about sets of small doubling.

I will discuss the general theory of Meyer sets and state Lagarias theorem. I will explain how additive combinatorics can be used to extend Lagarias result to discrete subsets of amenable groups. Going beyond the framework of amenable groups, I will talk about how one can use simple ideas from additive combinatorics in combination with powerful tools from ergodic theory - such as Zimmer’s cocycle superrigidity - to generalise Lagarias theorem to discrete subsets of SL_n(\mathbb{R}) for n > 2.