Duke Mathematical Journal

Near optimal bounds in Freiman's theorem

Tomasz Schoen

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove that if for a finite set $A$ of integers we have $|A+A|\le K|A|$, then $A$ is contained in a generalized arithmetic progression of dimension at most $K^{1+C(\log K)^{-1/2}}$ and of size at most $ {\rm exp}({K^{1+C(\log K)^{-1/2}}})|A|$ for some absolute constant $C$. We also discuss a number of applications of this result.

Article information

Source
Duke Math. J. Volume 158, Number 1 (2011), 1-12.

Dates
First available in Project Euclid: 3 May 2011

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1304429491

Digital Object Identifier
doi:10.1215/00127094-1276283

Zentralblatt MATH identifier
05904309

Mathematical Reviews number (MathSciNet)
MR2794366

Subjects
Primary: 11P70: Inverse problems of additive number theory, including sumsets
Secondary: 11B25: Arithmetic progressions [See also 11N13]

Citation

Schoen, Tomasz. Near optimal bounds in Freiman's theorem. Duke Mathematical Journal 158 (2011), no. 1, 1--12. doi:10.1215/00127094-1276283. http://projecteuclid.org/euclid.dmj/1304429491.


Export citation

References

  • Y. Bilu, “Structure of sets with small sumset” in Structure Theory of Set Addition, Astérisque 258, Soc. Math. France, Paris, 1999, 77–108.
  • E. Bombieri and U. Zannier, A note on squares in arithmetic progressions, II, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13 (2002), 69–75.
  • M.-C. Chang, A polynomial bound in Freiman's theorem, Duke Math. J. 113 (2002), 399–419.
  • —, On problems of Erdős and Rudin, J. Funct. Anal. 207 (2004), 444–460.
  • —, On sum-product representations in ${\mathbb {Z}}_q$, J. Eur. Math. Soc. (JEM) 8 (2006), 435–463.
  • —, Some consequences of the polynomial Freiman-Ruzsa conjecture, C. R. Math. Acad. Sci. Paris 347 (2009), 583–588.
  • K. Cwalina and T. Schoen, Linear bound on dimension in Green-Ruzsa's theorem, in preparation.
  • G. A. Freĭman, Foundations of a Structural Theory of Set Addition, Trans. Math. Monog. 37, Amer. Math. Soc., Providence, 1973.
  • W. T. Gowers, A new proof of Szemerédi's theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529–551.
  • —, “Rough structure and classification” in GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, 79–117.
  • —, A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001), 465–588.
  • B. Green, “Finite field models in additive combinatorics” in Surveys in Combinatorics 2005, London Math. Soc. Lecture Note Ser. 327, Cambridge Univ. Press, Cambridge, 2005, 1–27.
  • B. Green and I. Z. Ruzsa, Freiman's theorem in an arbitrary abelian group, J. Lond. Math. Soc. (2) 75 (2007), 163–175.
  • B. Green and T. Tao, An equivalence between inverse sumset theorems and inverse conjectures for the U$^3$ norm, Math. Proc. Cambridge. Philos. Soc. 149 (2010), 1–19.
  • N. H. Katz and P. Koester, On additive doubling and energy, preprint.
  • S. Konyagin and I. łaba, Distance sets of well-distributed planar sets for polygonal norms, Israel J. Math. 152 (2006), 157–175.
  • T. łuczak and T. Schoen, On a problem of Konyagin, Acta Arith. 134 (2008), 101–109.
  • I. Z. Ruzsa, Arithmetical progressions and the number of sums, Period. Math. Hungar. 25 (1992), 105–111.
  • —, Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65 (1994), 379–388.
  • T. Sanders, Additive structures in sumsets, Math. Proc. Cambridge. Philos. Soc. 144 (2008), 289–316.
  • —, Appendix to Roth's theorem on progressions revisited by J. Bourgain, J. Anal. Math. 104 (2008), 193–206.
  • —, On a nonabelian Balog-Szemerédi-type lemma, J. Aust. Math. Soc. 89 (2010), 127–132.
  • —, Structure in sets with logarithmic doubling, to appear in Canad. Math. Bull., preprint.
  • I. Shkredov and S. Yekhanin, Sets with large additive energy and symmetric sets, J. Combin. Theory Ser. A 118 (2011), 1086–1093.
  • B. Sudakov, E. Szemerédi, and V. H. Vu, On a question of Erdős and Moser, Duke Math. J. 129 (2005), 129–155.
  • E. Szemerédi and V. H. Vu, Long arithmetic progressions in sum-sets and the number of $x$-sum-free sets, Proc. Lond. Math. Soc. (3) 90 (2005), 273–296.
  • —, Finite and infinite arithmetic progressions in sumsets, Ann. of Math. (2) 163 (2006), 1–35.
  • —, Long arithmetic progressions in sumsets: Thresholds and bounds, J. Amer. Math. Soc. 19 (2006), 119–169.