Séminaire CAESAR
de combinatoire additive
Séance du jeudi 3 avril 2014
(14 heures 15, UPMC Paris 6 Jussieu, couloir 15-16, salle 101):
Matthew TOINTON
(Cambridge, G.B.)
Freiman's theorem in nilpotent group
Given a finite set A of integers, define A+A := { x + y : x, y in A
}. A classical theorem of Freiman classifies the finite sets A of integers that are "approximately closed", in the sense that A+A is "not too much bigger than" A. More precisely, Freiman's theorem states that if |A+A| < K|A| for some parameter K>1, then A must resemble something like an arithmetic progression; moreover, how closely A "resembles" an arithmetic progression is quantified in terms of K in a certain precise sense.
It makes sense to ask an analogous question in a more general setting: if A is a finite set in an arbitrary group, and we write AA := { xy : x, y in A }, then what can we say about A under the assumption that |AA| < K|A|? In recent years, there has been much progress on the answer to this more general question, and many interesting applications.
This talk will be in two parts. In the first part of the talk I will give a brief overview of what is known, and also how it has been useful elsewhere in mathematics. In the second half of the talk I will explain in more detail how the statement for the integers can be generalised to the case in which A is a subset of a nilpotent group.
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