The structure of approximate groups by Emmanuel Breuillard and others 2026-11-05
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Theorem 2.10 gives the approximate-group structure theorem. Corollary 11.9 and its proof, pages 68–69 in the inspected author version, obtain a bounded-index finite-by-nilpotent structure from one-scale relative polynomial growth. The proof uses a smaller scale with bounded growth; fixed scale constants may be adjusted by increasing the threshold.
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