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Abstract: Freiman-s theorem asserts, roughly speaking, if that a finite set in atorsion-free abelian group has small doubling, then it can be efficientlycontained in or controlled by a generalised arithmetic progression. This wasgeneralised by Green and Ruzsa to arbitrary abelian groups, where thecontrolling object is now a coset progression. We extend these results furtherto solvable groups of bounded derived length, in which the coset progressionsare replaced by the more complicated notion of a -coset nilprogression-. As oneconsequence of this result, any subset of such a solvable group of smalldoubling is is controlled by a set whose iterated products grow polynomially,and which are contained inside a virtually nilpotent group. As anotherapplication we establish a strengthening of the Milnor-Wolf theorem that allsolvable groups of polynomial growth are virtually nilpotent, in which only onelarge ball needs to be of polynomial size. This result complements recent workof Breulliard-Green, Fisher-Katz-Peng, and Sanders.

Autor: Terence Tao


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