Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

An equivalence between inverse sumset theorems and inverse conjectures for the U3 norm


a1 Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA. e-mail:

a2 UCLA Department of Mathematics, Los Angeles, CA 90095-1555, U.S.A. e-mail:


We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorems of Freĭman type are equivalent to the known inverse results for the Gowers U3 norms, and moreover that the conjectured polynomial strengthening of the former is also equivalent to the polynomial strengthening of the latter. We establish this equivalence in two model settings, namely that of the finite field vector spaces 2n, and of the cyclic groups xs2124/Nxs2124.

In both cases the argument involves clarifying the structure of certain types of approximate homomorphism.

(Received June 17 2009)

(Online publication March 24 2010)


† The first author holds a Leverhulme Prize and is grateful to the Leverhulme Trust for their support.

‡ The second author is supported by a grant from the MacArthur Foundation, and by NSF grant DMS-0649473.