... 1025–1054 A quantitative version of the idempotent theorem in harmonic analysis By Ben Green* and Tom SandersAbstractSuppose that G is a locally compact abelian group, and write M(G) for the algebra ... Balog-Szemer´edi-Gowers-Freiman). Let A be a subset of an abelian group G, and suppose that there are at least δ |A| 3additivequadruples (a 1, a 2, a 3, a 4) in A 4with a 1+ a 2= a 3+ a 4. Then there is a regularBourgain ... refinement of) Ruzsa’s analogue of Freiman’s theorem, which gives a fairly strong characterisation of subsets A ⊆Fn2satisfying a small doubling condition |A + A| K |A| . An analogue of this theorem...