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

Published online by Cambridge University Press:  24 March 2010

BEN GREEN  and
TERENCE TAO
BEN GREEN
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA. e-mail: [email protected]
TERENCE TAO
Affiliation:
UCLA Department of Mathematics, Los Angeles, CA 90095-1555, U.S.A. e-mail: [email protected]
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Abstract

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 ℤ/Nℤ.

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

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 1 , July 2010 , pp. 1 - 19
Copyright
Copyright © Cambridge Philosophical Society 2010

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