Equivalence of polynomial conjectures in additive combinatorics

Abstract

We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for U 3, which relates to functions which locally look like quadratics. In both cases a weak form, with exponential decay of parameters is known, and a strong form with only a polynomial loss of parameters is conjectured. Our main result is that the two conjectures are in fact equivalent.

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Correspondence to Shachar Lovett.

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Research supported by the Israel Science Foundation (grant 1300/05).

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Lovett, S. Equivalence of polynomial conjectures in additive combinatorics. Combinatorica 32, 607–618 (2012). https://doi.org/10.1007/s00493-012-2714-z

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Mathematics Subject Classification (2000)

  • 11P70