Finite approximate subrings in general rings admit a structure theorem where nilpotent quotients obstruct additive and multiplicative growth, yielding a general sum-product framework and a ring-theoretic analogue of Gromov's polynomial growth theorem.
Approximate groups [according to Hrushovski and Breuillard, Green, Tao]
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On the structure of approximate rings
Finite approximate subrings in general rings admit a structure theorem where nilpotent quotients obstruct additive and multiplicative growth, yielding a general sum-product framework and a ring-theoretic analogue of Gromov's polynomial growth theorem.