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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Balog and E. Szemerédi: A statistical theorem of set addition, Combinatorica, 14 (1994), 263–268.
V. Bergelson, T. Tao and T. Ziegler: An inverse theorem for the uniformity seminorms associated with the action of \(\mathcal{F}^\omega\), submitted, 2009.
G. A. Freiman: Foundations of a structural theory of set addition, by G. A. Freiman, American Mathematical Society, Providence, R.I., 1973.
W. T. Gowers: A new proof of Szemerédi’s theorem for arithmetic progressions of length four, GAFA, Geom. funct. anal. 8 (1998), 529–551.
B. Green and I. Z. Ruzsa: Sets with small sumset and rectification, Bulletin of the London Mathematical Society 38 (2006), 43–52.
B. Green: Finite field models in additive combinatorics, London Math. Soc. Lecture Note Ser., 327 (2005), 1–27.
B. Green and T. Tao: The distribution of polynomials over finite fields, with applications to the Gowers norms, submitted, 2007.
B. Green and T. Tao: An inverse theorem for the Gowers U 3 norm, Proc. Edinburgh Math. Soc. 51 (2008), 73–153.
B. Green and T. Tao: A note on the Freiman and Balog-Szemerédi-Gowers theorems in finite fields, J. Aust. Math. Soc., 86 (2009), 61–74.
B. Green and T. Tao: An equivalence between inverse sumset theorems and inverse conjectures for the U 3-norm, Math. Proc. Camb. Phil. Soc. 149 (2010), 1–19.
B. Green and T. Tao: Freiman’s theorem in finite fields via extremal set theory, Comb. Probab. Comput. 18 (2009), 335–355.
S. Lovett, R. Meshulam and A. Samorodnitsky: Inverse conjecture for the Gowers norm is false, In: Proceedings of the 40 th annual ACM symposium on Theory of computing (STOC’ 08), 547–556, New York, NY, USA, 2008. ACM.
I. Z. Ruzsa: An analog of Freiman’s theorem in groups, Structure Theory of Set-Addition, Astérique 258 (1999), 323–326.
A. Samorodnitsky: Low-degree tests at large distances, In: Proceedings of the 39th annual ACM symposium on Theory of computing (STOC’ 07), 506–515, New York, NY, USA, 2007. ACM.
T. Sanders: A note on Freiman’s theorem in vector spaces, Comb. Probab. Comput. 17 (2008), 297–305.
T. Tao and T. Ziegler: The inverse conjecture for the Gowers norm over finite fields via the correspondence principle, Analysis and PDE 3 (2010), 1–20.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by the Israel Science Foundation (grant 1300/05).
Rights and permissions
About this article
Cite this article
Lovett, S. Equivalence of polynomial conjectures in additive combinatorics. Combinatorica 32, 607–618 (2012). https://doi.org/10.1007/s00493-012-2714-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-012-2714-z