An approximate group is a symmetric subset P of a given group G which is “almost” closed under multiplication in the sense that the product set P^2 is contained in a finite union of translates of P. Over the years, this weakened – and perhaps seemingly artificial – extension of the notion of a group has naturally (and independently) appeared in many different areas in mathematics.
I will begin the talk by surveying some of these appearences (ranging from the sum-product phenomena of Erdos to Meyer’s quasi-crystals), hopefully convincing you that these appearances are not forced.
During the last few years I have, in joint collaboration with Tobias Hartnick (Technion), spent a non-trivial amount of time to try to better understand approximate groups which are “large” and “discrete” in some ambient locally compact group G (we call these subsets approximate lattices). During the second half of the talk I will try to motivate this course of research and survey some of our findings.
No previous knowledge of approximate groups will be required.