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

  • BEN GREEN (a1) and TERENCE TAO (a2)

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.

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[1] V. Bergelson , B. Host and B. Kra Multiple recurrence and nilsequences (with an appendix by I. Z. Ruzsa). Invent. Math. 160, 2 (2005), 261303.

[2] V. Bergelson and A. Leibman Distribution of values of bounded generalized polynomials. Acta Math. 198 (2007), 155230.

[4] M.–C. Chang A polynomial bound in Freĭman's theorem. Duke Math. J. 113 (2002), no. 3, 399419.

[12] B. J. Green and T. C. Tao Quadratic uniformity of the Möbius function. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 18631935.

[19] I. Z. Ruzsa Arithmetical progressions and the number of sums. Period. Math. Hungar. 25 (1992), no. 1, 105111.

[20] I. Z. Ruzsa Generalized arithmetical progressions and sumsets. Acta Math. Hungar. 65 (1994), no. 4, 379388.

[24] T. C. Tao Product set estimates for non-commutative groups. Combinatorica 28 (2008), no. 5, 547594.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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