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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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References

REFERENCES

[1]Bergelson, V., Host, B. and Kra, B.Multiple recurrence and nilsequences (with an appendix by I. Z. Ruzsa). Invent. Math. 160, 2 (2005), 261–303.CrossRefGoogle Scholar
[2]Bergelson, V. and Leibman, A.Distribution of values of bounded generalized polynomials. Acta Math. 198 (2007), 155–230.CrossRefGoogle Scholar
[3]Bergelson, V., Tao, T. C. and Ziegler, T. An inverse theorem for the uniformity seminorms associated with the action of F ω, to appear in GAFA. Available at arxiv.org/abs/0901.2602.Google Scholar
[4]Chang, M.–C.A polynomial bound in Freĭman's theorem. Duke Math. J. 113 (2002), no. 3, 399–419.CrossRefGoogle Scholar
[5]Davenport, H.Multiplicative number theory. Graduate Texts in Math. 74 (Springer, 3rd Ed, 2000).Google Scholar
[6]Freĭman, G. Foundations of a structural theory of set addition. Translated from the Russian. Translations of Mathematical Monographs, Vol. 37 (American Mathematical Society, 1973), vii + 108 pp.Google Scholar
[7]Gowers, W. T.A new proof of Szemerédi's theorem for arithmetic progressions of length four. GAFA 8 (1998), 529–551.Google Scholar
[8]Gowers, W. T.A new proof of Szemerédi's theorem. GAFA 11 (2001), 465–588.Google Scholar
[9]Gowers, W. T. Rough structure and classification. GAFA (2000) (Tel Aviv, 1999). Geom. Funct. Anal. (2000), Special Volume, Part I, 79–117.Google Scholar
[10]Green, B. J. Finite field models in additive combinatorics. Surveys in Combinatorics (2005), London Math. Soc. Lecture Notes 327, 1–27.Google Scholar
[11]Green, B. J. and Ruzsa, I. Z.Sets with small sumsets and rectification. Bull. London Math. Soc. 38 (2006), no. 1, 43–52.CrossRefGoogle Scholar
[12]Green, B. J. and Tao, T. C.Quadratic uniformity of the Möbius function. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 1863–1935.Google Scholar
[13]Green, B. J. and Tao, T. C.An inverse theorem for the Gowers U 3(G)-norm, with applications. Proc. Edinburgh Math. Soc. 51, no. 1, 73–153.CrossRefGoogle Scholar
[14]Green, B. J. and Tao, T. C. The distribution of polynomials over finite fields, with applications to the Gowers norms. To appear in Contrib. Discrete Math.Google Scholar
[15]Green, B. J. and Tao, T. C. Linear equations in primes. To appear in Ann. Math.Google Scholar
[16]Green, B. J. and Tao, T. C. The quantitative behaviour of polynomial orbits on nilmanifolds. Preprint available at arxiv.org/abs/0709.3562.Google Scholar
[17]Host, B. and Kra, B. Analysis of two step nilsequences. Preprint available at arxiv.org/abs/0709.3241.Google Scholar
[18]Lovett, S., Meshulam, R. and Samorodnitsky, A. Inverse Conjecture for the Gowers norm is false. Preprint available at arxiv.org/abs/0711.3388.Google Scholar
[19]Ruzsa, I. Z.Arithmetical progressions and the number of sums. Period. Math. Hungar. 25 (1992), no. 1, 105–111.CrossRefGoogle Scholar
[20]Ruzsa, I. Z.Generalized arithmetical progressions and sumsets. Acta Math. Hungar. 65 (1994), no. 4, 379–388.CrossRefGoogle Scholar
[21]Ruzsa, I. Z.Sums of finite sets, Number Theory: New York Seminar (Springer-Verlag, 1996), Edited by Chudnovsky, D. V., Chudnovsky, G. V. and Nathanson, M. B.Google Scholar
[22]Ruzsa, I. Z.An analog of Freĭman's theorem in groups. Structure theory of set addition. Astérisque 258 (1999), 323–326.Google Scholar
[23]Samorodnitsky, A. Low-degree tests at large distances. STOC (2007).CrossRefGoogle Scholar
[24]Tao, T. C.Product set estimates for non-commutative groups. Combinatorica 28 (2008), no. 5, 547–594.CrossRefGoogle Scholar
[25]Tao, T. C. Freĭman's theorem for solvable groups. Preprint.Google Scholar
[26]Tao, T. C. and Vu, V. H. Additive Combinatorics. Cambridge Studies in Advanced Math. 105, (Cambridge University Press, 2006).CrossRefGoogle Scholar
[27]Tao, T. C. and Ziegler, T. The inverse conjecture for the Gowers norm over finite fields via the correspondence principle. Preprint available at arxiv.org/abs/0810.5527.Google Scholar