The structure of approximate groups

@article{Breuillard2011TheSO,
  title={The structure of approximate groups},
  author={Emmanuel Breuillard and Ben Green and Terence Tao},
  journal={Publications math{\'e}matiques de l'IH{\'E}S},
  year={2011},
  volume={116},
  pages={115-221}
}
Let K⩾1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A⋅A is covered by K left translates of A.The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups.We… 
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