Type: Article
Publication Date: 2007-01-29
Citations: 188
DOI: https://doi.org/10.1112/jlms/jdl021
Journal of the London Mathematical SocietyVolume 75, Issue 1 p. 163-175 Articles Freiman's theorem in an arbitrary abelian group Ben Green, Ben Green Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom, [email protected]Search for more papers by this authorImre Z. Ruzsa, Imre Z. Ruzsa Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, Budapest, Pf. 127, H-1364, Hungary, [email protected]Search for more papers by this author Ben Green, Ben Green Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom, [email protected]Search for more papers by this authorImre Z. Ruzsa, Imre Z. Ruzsa Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, Budapest, Pf. 127, H-1364, Hungary, [email protected]Search for more papers by this author First published: 29 January 2007 https://doi.org/10.1112/jlms/jdl021Citations: 70 2000 Mathematics Subject Classification 11P70, 11B99. While this work was carried out, the first author was supported by a PIMS postdoctoral fellowship at the University of British Columbia, Vancouver, Canada. Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat Abstract A famous result of Freiman describes the structure of finite sets A ⊆ ℤ with small doubling property. If |A + A| ⩽ K|A|, then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)|A|. Here we prove an analogous statement valid for subsets of an arbitrary abelian group. References 1.Y. Bilu, Structure of sets with small sumset, Structure theory of set addition, 1999, Paris, Société Mathématique de France, 77–108, Astérisque 258 2.N. N. Bogolyubov, Sur quelques propriétés arithmétiques des presque-périodes, Ann. Chaire Math. Phys. Kiev, 1939, 4, 185–194 3.J. W. S. Cassels, An introduction to the geometry of numbers, Classics in mathematics, 1997, Berlin, Springer, corrected reprint of the 1971 edition 4.M. C. Chang, A polynomial bound in Freiman's theorem, Duke Math. J., 2002, 113, 399–419 5.J.-M. Deshouillers, F. Hennecart, A. Plagne, On small sumsets in (ℤ/2ℤ)n, Combinatorica, 2004, 24, 53–68 6.G. Freiman, Foundations of a structural theory of set addition, 1973, 37, Providence, RI, American Mathematical Society, Translations of Mathematical Monographs 7.B. J. Green, Edinburgh–MIT lecture notes on Freiman's theorem, 2006, Preprint, http://www.dpmms.cam.ac.uk/~bjg23 8.B. J. Ruzsa, I. Z. Green, Sets with small sumset and rectification, Bull. London Math. Soc., 2006, 38, 43–52 9.H. Plünnecke, Eigenschaften un Abschätzungen von Wirkingsfunktionen, 1969, Bonn, Gesellschaft für Mathematik und Datenverarbeitung, BMwF-GMD-22 10.W. Rudin, Interscience Tracts in Pure and Applied Mathematics 12, Fourier analysis on groups, 1962, New York, Wiley-Interscience 11.I. Z. Ruzsa, On the cardinality of A + A and A − A, 1978, North-Holland, 933–938, Keszthely, 1976, Vol. II, Colloquia Mathematica Societatis János Bolyai 18 12.I. Z. Ruzsa, An application of graph theory to additive number theory, Sci. Ser. A Math. Sci. (N.S.), 1989, 3, 97–109 13.I. Z. Ruzsa, Generalized arithmetical progressions and sumsets, Acta Math. Hungar., 1994, 65, 379–388 14.I. Z. Ruzsa, An analog of Freiman's theorem in groups, Structure theory of set addition, 1999, Paris, Société Mathématique de France, 323–326, Astérisque 258 Citing Literature Volume75, Issue1February 2007Pages 163-175 ReferencesRelatedInformation