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Published online by Cambridge University Press: 31 October 2019
We state Breuillard, Green and Tao’s rough classification of the finite approximate subgroups of an arbitrary group. This states that a finite approximate subgroup of an arbitrary group is contained in a union of a few cosets of a finite-by-nilpotent group, the nilpotent quotient of which has bounded step. We define coset nilprogressions, and show how to deduce a more detailed version of the Breuillard–Green–Tao theorem in which the approximate subgroup is contained in a union of a few translates of a coset nilprogression of bounded rank and step.
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