Small Doubling in Groups

  • Emmanuel Breuillard
  • Ben Green
  • Terence Tao
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 25)


Let A be a subset of a group G = (G; ·). We will survey the theory of sets A with the property that |A · A|≤K|A|, where A · A = {a1a2: a1; a2 ∈ A}. The case G = (ℤ; +) is the famous Freiman-Ruzsa theorem.


Cayley Graph Polynomial Growth Matrix Group Abelian Case Expander Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Y. Bilu. Structure of sets with small sumset. Astérisque, (258): xi, 77–108, 1999. Structure theory of set addition.Google Scholar
  2. [2]
    J. Bourgain and A. Gamburd. Uniform expansion bounds for Cayley graphs of SL2 (147-1). Ann. of Math. (2), 167(2):625–642, 2008.Google Scholar
  3. [3]
    J. Bourgain, N. Katz, and T. Tao. A sum-product estimate in finite fields, and applications. Geom. Funct. Anal., 14(1):27–57, 2004.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    E. Breuillard. A mini-course on approximate groups. to appear in Proceedings of the Mathematical Sciences Research Institute (MSRI), 2013. Preprint.Google Scholar
  5. [5]
    E. Breuillard and B. Green. Approximate groups. I: the torsion-free nilpotent case. J. Inst. Math. Jussieu, 10(1):37–57, 2011.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    E. Breuillard and B. Green. Approximate groups, II: The solvable linear case. Q.J. Math., 62(3):513–521, 2011.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    E. Breuillard, B. Green, and T. Tao. Approximate subgroups of linear groups. Geom. Funct. Anal., 21(4):774–819, 2011.CrossRefzbMATHMathSciNetGoogle Scholar
  8. nr][9]
    E. Breuillard, B. Green, and T. Tao. A note on approximate subgroups of GLn (ℂ) and uniformly nonamenable groups. arXiv:1101.2552, 2011. Preprint.Google Scholar
  9. [10]
    E. Breuillard, B. Green, and T. Tao. The structure of approximate groups. arXiv:1110.5008, 2011. Preprint.Google Scholar
  10. [11]
    A. L. Cauchy. Recherches sur les nombres. J. École Polytech., 9:99–116, 1813.Google Scholar
  11. [12]
    M.-C. Chang. A polynomial bound in Freiman’s theorem. Duke Math. J., 113(3): 399–419, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [13]
    M.-C. Chang. Product theorems in SL2 and SL3. J. Inst. Math. Jussieu, 7(1):1–25, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [14]
    T. Coulhon, L. Saloff-Coste, and N. T. Varopoulos. Analysis and geometry on groups, volume 100 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1992.Google Scholar
  14. [15]
    E. Croot and O. Sisask. A probabilistic technique for finding almost-periods of convolutions. Geom. Funct. Anal., 20(6):1367–1396, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [16]
    H. Davenport. On the addition of residue classes. J. London Math. Soc., 10:30–32, 1935.Google Scholar
  16. [17]
    M. DeVos. The structure of critical product sets. arXiv:1301.0096, 2013. Preprint.Google Scholar
  17. [18]
    O. Dinai. Expansion properties of finite simple groups. arXiv:1001.5069, 2010. Preprint.Google Scholar
  18. [19]
    J. D. Dixon. The probability of generating the symmetric group. Math. Z., 110:199–205, 1969.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [20]
    G. Elekes and Z. Király. On the combinatorics of projective mappings. J. Algebraic Combin., 14(3):183–197, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [21]
    C. Even-Zohar. On sums of generating sets in (ℤ2)n. Combin. Probab. Comput., 21(6):916–941, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [22]
    D. Fisher, N. H. Katz, and I. Peng. Approximate multiplicative groups in nilpotent Lie groups. Proc. Amer. Math. Soc., 138(5):1575–1580, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [23]
    G. A. Freiman. Foundations of a structural theory of set addition. American Mathematical Society, Providence, R. I., 1973. Translated from the Russian, Translations of Mathematical Monographs, Vol 37.Google Scholar
  23. [24]
    G. A. Freiman. Groups and the inverse problems of additive number theory. In Number-theoretic studies in the Markov spectrum and in the structural theory of set addition (Russian), pages 175–183. Kalinin. Gos. Univ., Moscow, 1973.Google Scholar
  24. [25]
    G. A. Freiman. On finite subsets of nonabelian groups with small doubling. Proc. Amer. Math. Soc., 1406(9):2997–3002, 2012.CrossRefMathSciNetGoogle Scholar
  25. [26]
    R. J. Gardner. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.), 39(3):355–405, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [27]
    N. Gill and H. Helfgott. Growth in solvable subgroups of GLr(ℤ/pℤ). arXiv:1008.5264, 2010. Preprint.Google Scholar
  27. [28]
    A. S. Golsefidy and P. P. Varjú. Expansion in perfect groups. Geom. Funct. Anal., 22(6):1832–1891, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  28. [29]
    W. T. Gowers. A new proof of Szemerédi’s theorem. Geom. Funct. Anal., 11(3):465–588, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [30]
    W. T. Gowers. Quasirandom groups. Combin. Probab. Comput., 17(3):363–387, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [32]
    B. Green. Finite field models in additive combinatorics. In Surveys in combinatorics 2005, volume 327 of London Math. Soc. Lecture Note Ser., pages 1–27. Cambridge Univ. Press, Cambridge, 2005.Google Scholar
  31. [33]
    B. Green and I. Z. Ruzsa. Freiman’s theorem in an arbitrary abelian group. J. Lond. Math. Soc. (2), 75(1):163–175, 2007.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [34]
    B. Green and T. Tao. Compressions, convex geometry and the Freiman-Bilu theorem. Q.J. Math., 57(4):495–504, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [35]
    B. Green and T. Tao. Freiman’s theorem in finite fields via extremal set theory. Combin. Probab. Comput., 18(3):335–355, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [36]
    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(1):1–19, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  35. [37]
    R. Grigorchuk. On the Gap Conjecture concerning group growth., 2012.Google Scholar
  36. [38]
    R. I. Grigorchuk. On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen., 14(1):53–54, 1980.MathSciNetGoogle Scholar
  37. [39]
    R. I. Grigorchuk. On growth in group theory. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 325–338, Tokyo, 1991. Math. Soc. Japan.Google Scholar
  38. [40]
    M. Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Etudes Sci. Publ. Math., (53):53–73, 1981.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [41]
    Y. O. Hamidoune. Two inverse results. arXiv:1006.5074, 2010. Preprint.Google Scholar
  40. [42]
    H. A. Helfgott. Growth and generation in SL2(ℤ/pℤ). Ann. of Math. (2), 167(2): 601–623, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  41. [43]
    H. A. Helfgott. Growth in SL3(ℤ/pℤ). J. Eur. Math. Soc. (JEMS), 13(3):761–851, 2011.CrossRefzbMATHMathSciNetGoogle Scholar
  42. [44]
    S. Hoory, N. Linial, and A. Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4):439–561 (electronic), 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  43. [45]
    E. Hrushovski. Stable group theory and approximate subgroups. J. Amer. Math. Soc., 25(1):189–243, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [46]
    M. Kassabov. Symmetric groups and expanders. Electron. Res. Announc. Amer. Math. Soc., 11:47–56 (electronic), 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  45. [47]
    M. Kneser. Abschätzung der asymptotischen Dichte von Summenmengen. Math. Z., 58:459–484, 1953.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [48]
    S. V. Konyagin. On Freiman’s theorem in finite fields. Mat. Zametki, 84(3):472–474, 2008.CrossRefMathSciNetGoogle Scholar
  47. [49]
    M. J. Larsen and R. Pink. Finite subgroups of algebraic groups. J. Amer. Math. Soc., 24(4):1105–1158, 2011.CrossRefzbMATHMathSciNetGoogle Scholar
  48. [50]
    V. F. Lev and P. Y. Smeliansky. On addition of two distinct sets of integers. Acta Arith., 70(1):85–91, 1995.zbMATHMathSciNetGoogle Scholar
  49. [51]
    S. Lovett. Equivalence of polynomial conjectures in additive combinatorics. arXiv:1001.3356, 2010. Preprint.Google Scholar
  50. [52]
    A. Lubotzky. Expander graphs in pure and applied mathematics (notes for the colloquium lectures, AMS annual meeting 2001).Google Scholar
  51. [53]
    A. Lubotzky. Cayley graphs: eigenvalues, expanders and random walks. In Surveys in combinatorics, 1995 (Stirling), volume 218 of London Math. Soc. Lecture Note Ser., pages 155–189. Cambridge Univ. Press, Cambridge, 1995.CrossRefGoogle Scholar
  52. [54]
    A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261–277, 1988.CrossRefzbMATHMathSciNetGoogle Scholar
  53. [55]
    G. A. Margulis. Explicit constructions of expanders. Problemy Peredači Informacii, 9(4):71–80, 1973.zbMATHMathSciNetGoogle Scholar
  54. [56]
    G. Petridis. Plünnecke’s inequality. Combin. Probab. Comput., 20(6):921–938, 2011.CrossRefzbMATHMathSciNetGoogle Scholar
  55. [57]
    H. Plünnecke. Eigenschaften und Abschätzungen von Wirkungsfunktionen. BMwFGMD-22. Gesellschaft für Mathematik und Datenverarbeitung, Bonn, 1969.Google Scholar
  56. [58]
    L. Pyber and E. Szabó. Growth in finite simple groups of Lie type of bounded rank. arXiv:1005.1858, 2010. Preprint.Google Scholar
  57. [59]
    L. Pyber and E. Szabó. Growth in linear groups. to appear in Proceedings of the Mathematical Sciences Research Institute (MSRI), 2013. Preprint.Google Scholar
  58. [61]
    I. Z. Ruzsa. Generalized arithmetical progressions and sumsets. Acta Math. Hungar., 65(4):379–388, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  59. [62]
    I. Z. Ruzsa. Sums of finite sets. In Number theory (New York, 1991–1995), pages 281–293. Springer, New York, 1996.Google Scholar
  60. [63]
    I. Z. Ruzsa. An analog of Freiman’s theorem in groups. Astérisque, (258): xv, 323–326, 1999. Structure theory of set addition.Google Scholar
  61. [64]
    I. Z. Ruzsa. Sumsets and structure. In Combinatorial number theory and additive group theory, Adv. Courses Math. CRM Barcelona, pages 87–210. Birkhäuser Verlag, Basel, 2009.Google Scholar
  62. [65]
    S. R. Safin. Powers of subsets of free groups. Mat. Sb., 202(11):97–102, 2011.CrossRefMathSciNetGoogle Scholar
  63. [66]
    T. Sanders. On a nonabelian Balog-Szemerédi-type lemma. J. Aust. Math. Soc., 89(1):127–132, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  64. [67]
    T. Sanders. An analytic approach to a weak non-Abelian Kneser-type theorem. arXiv:1212.0457, 2012. Preprint.Google Scholar
  65. [68]
    T. Sanders. Approximate groups and doubling metrics. Math. Proc. Cambridge Philos. Soc., 152(3):385–404, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  66. [69]
    T. Sanders. On the Bogolyubov-Ruzsa lemma. Anal. PDE, 5(3):627–655, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  67. [70]
    T. Sanders. The structure theory of set addition revisited. Bull. Amer. Math. Soc. (N.S.), 50(1):93–127, 2013.CrossRefzbMATHMathSciNetGoogle Scholar
  68. [71]
    P. Sarnak. What is … an expander? Notices Amer. Math. Soc., 51(7):762–763, 2004.zbMATHMathSciNetGoogle Scholar
  69. [72]
    P. Sarnak and X. X. Xue. Bounds for multiplicities of automorphic representations. Duke Math. J., 64(1):207–227, 1991.CrossRefzbMATHMathSciNetGoogle Scholar
  70. [73]
    T. Schoen. Near optimal bounds in Freiman’s theorem. Duke Math. J., 158(1):1–12, 2011.CrossRefzbMATHMathSciNetGoogle Scholar
  71. [74]
    Y. Shalom and T. Tao. A finitary version of Gromov’s polynomial growth theorem. Geom. Funct. Anal., 20(6):1502–1547, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  72. [75]
    Y. Stanchescu. On the structure of sets with small doubling property on the plane. I. Acta Arith., 83(2):127–141, 1998.MathSciNetGoogle Scholar
  73. [76]
    Y. Stanchescu. On the structure of sets with small doubling property on the plane. II. Integers, 8(2):20, 2008.MathSciNetGoogle Scholar
  74. [79]
    T. Tao. Product set estimates for non-commutative groups. Combinatorica, 28(5): 547–594, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  75. [80]
    T. Tao. Freiman’s theorem for solvable groups. Contrib. Discrete Math., 5(2):137–184, 2010.MathSciNetGoogle Scholar
  76. [81]
    T. Tao. Noncommutative sets of small doubling. arXiv:1106.2267, 2011. Preprint.Google Scholar
  77. [82]
    T. Tao and V. Vu. Additive combinatorics, volume 105 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006.Google Scholar
  78. ai][83]
    M. Tointon. Freiman’s theorem in an arbitrary nilpotent group. arXiv:1211.3989, 2012. Preprint.Google Scholar
  79. [84]
    L. van den Dries and A. J. Wilkie. Gromov’s theorem on groups of polynomial growth and elementary logic. J. Algebra, 89(2):349–374, 1984.CrossRefzbMATHMathSciNetGoogle Scholar
  80. [85]
    P. P. Varjú. Expansion in SLd (O K/I), I square-free. J. Eur. Math. Soc. (JEMS), 14(1):273–305, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  81. [86]
    A. G. Vosper. The critical pairs of subsets of a group of prime order. J. London Math. Soc., 31:200–205, 1956.CrossRefzbMATHMathSciNetGoogle Scholar
  82. [87]
    W. Woess. Random walks on in_nite graphs and groups, volume 138 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2000.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2013

Authors and Affiliations

  • Emmanuel Breuillard
    • 1
  • Ben Green
    • 2
  • Terence Tao
    • 3
  1. 1.Laboratoire de MathématiquesUniversité Paris Sud 11OrsayFrance
  2. 2.Centre for Mathematical SciencesCambridgeEngland
  3. 3.Department of MathematicsUCLALos AngelesUSA

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