Abstract
We develop the Plünnecke-Ruzsa and Balog-Szemerédi-Gowers theory of sum set estimates in the non-commutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freiman-type inverse theorem for a special class of 2-step nilpotent groups, namely the Heisenberg groups with no 2-torsion in their centre.
Similar content being viewed by others
References
A. Balog and E. Szemerédi: A statistical theorem of set addition, Combinatorica 14(3) (1994), 263–268.
Y. Bilu: Structure of sets with small sumset, in: Structure theory of set addition, Astérisque No. 258 (1999), xi, 77–108.
J. Bourgain: On the dimension of Kakeya sets and related maximal inequalities, Geom. Func. Anal. 9 (1999), 256–282.
J. Bourgain: Estimates on exponential sums related to the Diffie-Hellman distributions, Geom. Funct. Anal. 15(1) (2005), 1–34.
J. Bourgain: Mordell’s exponential sum estimate revisited, J. Amer. Math. Soc. 18(2) (2005), 477–499.
J. Bourgain, N. Katz and T. Tao: A sum-product estimate in finite fields, and applications; Geom. Func. Anal. 14 (2004), 27–57.
J. Bourgain and S. Konyagin: Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order, C. R. Acad. Sci. Paris, Ser. I 337 (2003), 75–80.
L. V. Brailovsky and G. A. Freiman: On a product of finite subsets in a torsionfree group, J. Algebra 130 (1990), 462–476.
M. Chang: A polynomial bound in Freiman’s theorem, Duke Math. J. 113(3) (2002), 399–419.
M. C. Chang: On problems of Erdös and Rudin, J. Funct. Anal. 207 (2004), 444–460.
G. Elekes: On linear combinatorics I, Combinatorica 17(4) (1997), 447–458.
G. Elekes: On linear combinatorics II, Combinatorica 18(1) (1998), 13–25.
G. Elekes: On linear combinatorics III, Combinatorica 19(1) (1999), 43–53.
G. Elekes and Z. Király: On combinatorics of projective mappings, J. Alg. Combin. 14 (2001), 183–197.
G. Elekes and I. Z. Ruzsa: The structure of sets with few sums along a graph, J. Combin. Theory Ser. A 113(7) (2006), 1476–1500.
G. Freiman: Foundations of a structural theory of set addition (Translated from the Russian), Translations of Mathematical Monographs, Vol. 37, American Mathematical Society, Providence, R. I., 1973, vii+108 pp.
T. Gowers: A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geom. Func. Anal. 8 (1998), 529–551.
T. Gowers: A new proof of Szemeredi’s theorem, Geom. Func. Anal. 11 (2001), 465–588.
B. Green: Finite field models in arithmetic combinatorics, preprint.
B. Green and I. Z. Ruzsa: Freiman’s theorem in an arbitrary abelian group, J. London Math. Soc. 75(1) (2007), 163–175.
B. Green and T. Tao: Compressions, Convex Geometry and the Freiman-Bilu Theorem; Q. J. Math. 57(4) (2006), 495–504.
H. Helfgott: Growth and generation in SL2(Z/pZ), Ann. Math., accepted (2007). arXiv:math/0509024.
Y. Hamidoune, A. S. Lladó and O. Serra: On subsets with small product in torsion-free groups, Combinatorica 18(4) (1998), 529–540.
M. Laczkovich and I. Z. Ruzsa: The number of homothetic subsets, in: The Mathematics of Paul Erdős (Graham and Nešetřil eds.), Springer, 1996.
E. Lindenstrauss: Pointwise theorems for amenable groups, Invent. Math. 146(2) (2001), 259–295.
J. H. B. Kemperman: On complexes in a semigroup, Indag. Math. 18 (1956), 247–254.
V. Milman: Entropy and asymptotic geometry of non-symmetric convex bodies, Adv. in Math. 152 (2000), 314–335.
M. Nathanson: Additive number theory. Inverse problems and the geometry of sumsets, Graduate Texts in Mathematics 165, Springer-Verlag, New York, 1996.
H. Plünnecke: Eigenschaften und Abschätzungen von Wirkungsfunktionen, BMwFGMD-22 Gesellschaft für Mathematik und Datenverarbeitung, Bonn, 1969.
I. Z. Ruzsa: Sums of finite sets, in: Number Theory (D. V. Chudnovsky, G. V. Chudnovsky and M. B. Nathanson, editors), Springer-Verlag, New York, 1996, pp. 281–293.
I. Z. Ruzsa: Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65(4) (1994), 379–388.
I. Z. Ruzsa: An analog of Freiman’s theorem in groups, in: Structure theory of set addition, Astérisque No. 258 (1999), 323–326.
I. Z. Ruzsa and S. Turjányi: A note on additive bases of integers, Publ. Math. Debrecen 32 (1985), 101–104.
B. Sudakov, E. Szemerédi and V. H. Vu: On a question of Erd?os and Moser, Duke Math. J. 129(1) (2005), 129–155.
T. Tao: Non-commutative sum set estimates, unpublished.
T. Tao and V. H. Vu: Additive Combinatorics, Cambridge University Press, Cambridge, 2006, 530 pp.
Author information
Authors and Affiliations
Corresponding author
Additional information
T. Tao is supported by a grant from the Packard Foundation.
Rights and permissions
About this article
Cite this article
Tao, T. Product set estimates for non-commutative groups. Combinatorica 28, 547–594 (2008). https://doi.org/10.1007/s00493-008-2271-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-008-2271-7