Nilpotent approximate groups

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• Matthew Tointon (Cambridge)
• Wednesday 31 October 2012, 16:00-17:00
• MR11, CMS.

If you have a question about this talk, please contact Ben Green.

A fundamental theorem of additive combinatorics is Freiman’s theorem, proved in the 1960s, the statement of which is roughly as follows. Suppose a set A of integers has the property that the number of integers that can be expressed as the sum of two (not necessarily distinct) members of A is ‘not much greater’ than the cardinality of A. Then Freiman’s theorem says that A is contained inside a low-dimensional generalised arithmetic progression of cardinality not much greater than that of A. In recent years Green and Ruzsa generalised this result to all abelian groups, where the conclusion is that A is efficiently contained inside the sum of a finite subgroup and a low-dimensional progression.

In non-abelian groups the analogue of the hypothesis of Freiman’s theorem is that A is an ‘approximate group’, which roughly means that the set of all elements of the form xy, with x and y belonging to A, can be covered by a few translates of A. Generalising Freiman’s theorem in a different direction, Breuillard and Green have shown that approximate subgroups inside any torsion-free nilpotent group can be controlled by progressions, provided that the notion of progression is suitably modified from the abelian version.

In this talk I will give an outline of a result that generalises all of these statements. Specifically, this result says that an approximate subgroup A of an arbitrary nilpotent group G is efficiently contained in the product of a finite subgroup normalised by A and a nilpotent progression. I will probably specialise to the case in which G is of nilpotency class 2, in which the ideas of the argument are all present but the technical details are far simpler than in general. If there is time at the end I will give some pointers on how to proceed to the general case.

This talk is part of the Discrete Analysis Seminar series.

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