Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-07T21:47:22.909Z Has data issue: false hasContentIssue false
  • English
  • Français

A Step Beyond Kemperman's Structure Theorem

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  21 December 2009

David J. Grynkiewicz
David J. Grynkiewicz
Affiliation:
Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstrasse 36, 8010 GrazHeinrichstrasse 36, 8010 Graz, Austria, E-mail: diambri@hotmail.com
Get access

Abstract

We extend Kemperman's structure theorem by completely characterizing those finite subsets A and B of an arbitrary abelian group with |A + B| = |A| + |B|.

MSC classification

Secondary: 11P70: Inverse problems of additive number theory, including sumsets
Type
Research Article
Information
Mathematika , Volume 55 , Issue 1-2 , December 2009 , pp. 67 - 114
Copyright
Copyright © University College London 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bilu, Y. F., Lev, V. F. and Ruzsa, I. Z., Rectification principles in additive number theory, dedicated to the memory of Paul Erős. Discrete Comput. Geom. 19(3) (1998), 343–353 (special issue).CrossRefGoogle Scholar
2.Chowla, S., A theorem on the addition of residue classes: applications to the number г(k) in Warings problem. Proc. Indian Acad. Sci. Math. Sci. 2 (1935), 242–243.CrossRefGoogle Scholar
3.Deshouillers, J. and Freiman, G. A., A step beyond Kneser's theorem for abelian finite groups. Proc. Lond. Math. Soc. (3) 86(1) (2003), 1–28.CrossRefGoogle Scholar
4.Freiman, G. A., The addition of finite sets I. Izv. Vyssh. Uchebn. Zaved. Mat. 6(13) (1959), 202–213 (in Russian).Google Scholar
5.Freiman, G. A., Inverse problems in additive number theory. Addition of sets of residues modulo a prime. Dokl. Akad. Nauk SSSR 141 (1961), 571–573 (in Russian).Google Scholar
6.Freiman, G. A., Inverse problems of additive number theory. On the addition of sets of residues with respect to a prime modulus. Soviet Math.-Dokl. 2 (1961), 1520–1522.Google Scholar
7.Freiman, G. A., Inverse problems of additive number theory VI: on the addition of finite sets III. Izv. Vyssh. Uchebn. Zaved. Mat. 3(28) (1962), 151–157 (in Russian).Google Scholar
8.Freiman, G. A., Foundations of a Structural Theory of Set Addition (Translations of Mathematical Monographs 37), American Mathematical Society (Providence, RI, 1973) (translated from the Russian).Google Scholar
9.Grynkiewicz, D. J., An extension of the Erdős-Ginzburg-Ziv theorem to hypergraphs. European J. Combin. 26(8) (2005), 1154–1176.CrossRefGoogle Scholar
10.Grynkiewicz, D. J., Quasi-periodic decompositions and the Kemperman structure theorem. European J. Combin. 26(5) (2005), 559–575.CrossRefGoogle Scholar
11.Grynkiewicz, D. J., Sumsets, zero-sums and extremal combinatorics. PhD Dissertation, Department of Mathematics, California Institute of Technology, 2006.Google Scholar
12.Grynkiewicz, D. J., On extending Pollard's theorem for t-representable sums. Israel J. Math. (to appear).Google Scholar
13.Grynkiewicz, D. J., Lev, V. and Serra, O., Connectivity of addition Cayley graphs. J. Combin. Theory, Ser. B 99 (2009), 202–217.CrossRefGoogle Scholar
14.Grynkiewicz, D. J., Marchan, E. and Ordaz, O., Representation of finite abelian group elements by subsequence sums. J. Théor. Nombres Bordeaux (to appear).Google Scholar
15.Grynkiewicz, D. J. and Serra, O., The Freiman 3k theorem: distinct summands(in preparation).Google Scholar
16.Hamidoune, Y. O., Hyper-atoms and the critical pair theory Preprint.Google Scholar
17.Hamidoune, Y. O., Hyper-atoms and the Kemperman's critical pair theory. Preprint.Google Scholar
18.Hamidoune, Y. O., Subsets with small sums in abelian groups I: the Vosper property. European J. Combin. 18(5) (1997), 541–556.CrossRefGoogle Scholar
19.Hamidoune, Y. O., Subsets with a small sum II: the critical pair problem. European J. Combin. J. 21(2) (2000), 231–239.CrossRefGoogle Scholar
20.Hamidoune, Y. O., Lladó, A. S. and Serra, O., Vosperian and superconnected abelian Cayley digraphs. Graphs Combin. 7(2) (1991), 143–152.CrossRefGoogle Scholar
21.Hamidounee, Y. O., Montejano, A. and Serra, O.., Rainbow-free three colorings in abelian groups. Preprint.Google Scholar
22.Hamidoune, Y. O. and Rødseth, J. O., An inverse theorem mod p. Acta. Arith. 92(3) (2000), 251–262.CrossRefGoogle Scholar
23.Hamidoune, Y. O., and Serra, O. and Zemor, G.., On the critical pair theory in ℤ/pℤ. Acta Arith. 121(2) (2006).CrossRefGoogle Scholar
24.Hamidoune, Y. O., Serra, O. and Zemor, G.., On the critical pair theory in abelian groups: beyond Chowla's theorem. Combinatorica 28(4) (2008), 441–467.CrossRefGoogle Scholar
25.Jungć, V.Nešetřil, J. and Radoičić, R., Rainbow Ramsey Theory. Integers 5(2) (2005), A9 (electronic)Google Scholar
26.Gárolyi, G. K., The Erdäs-Heilbronn problem in abelian groups. Israel J. Math.. 139 (2004), 349–359.Google Scholar
27.Gárolyi, G. K., An inverse theorem for the restricted set addition in abelian groups. J. Algebra 290(2) (2005), 557–593.Google Scholar
28.Kemperman, J. H. B., On small sumsets in an abelian group. Acta Math. 103 (1960), 63–88.CrossRefGoogle Scholar
29.Kneser, M., Abschätzung der asymptotischen dichte von summenmengen. Math. Z. 58 (1953), 459–484.CrossRefGoogle Scholar
30.Kneser, M., Ein satz über abelsche gruppen mit anwendungen auf die geometrie der zahlen. Math. Z. 64 (1955), 429–434.Google Scholar
31.Kneser, M., Summenmengen in lokalkompakten abelschen gruppen. Math. Z. 66 (1956), 8–110.CrossRefGoogle Scholar
32.Lev, V., On small sumsets in abelian groups. Astérisque 258 (1999), 317–321.Google Scholar
33.Lev, V.., Critical pairs in abelian groups and Kemperman's structure theorem. Int. J. Number Theory 2(3) (2006), 379–396.CrossRefGoogle Scholar
34.Nathanson, M. B., Additive Number Theory: Inverse Problems and the Geometry of Sumsets (Graduate Texts in Mathematics 165), Springer (New York, 1996).CrossRefGoogle Scholar
35.Pollard, J. M., A Generalisation of the Theorem of Cauchy and Davenport. J. Davenport. Lond. Math. Soc. (2) 8 (1974), 460–462.CrossRefGoogle Scholar
36.Serra, O., An Isoperimetric Method for the Small Sumset Problem. In Surveys in Combinatorics 2005 (London Mathematical Society Lecture Note Series 327) (ed. Webb, B. S.), Cambridge University Press (Cambridge, 2005), 119–152.CrossRefGoogle Scholar
37.Serra, O. and Gémor, G. Z., On a generalization of a theorem by Vosper. Integers (2000) (electronic).Google Scholar
38.Tao, T. and Vu, V., Additive Combinatorics (Cambridge Studies in Advanced Mathematics 105), Cambridge University Press (Cambridge, 2006).CrossRefGoogle Scholar
39.Vosper, A. G., The critical pairs of subsets of a group of prime order. J. order. Lond. Math. Soc. 31 (1956), 200–205.CrossRefGoogle Scholar
40.Vosper, A. G., Addendum to “The critical pairs of subsets of a group of prime order”. J. Lond. Math. Soc. 31 (1956), 280–282.CrossRefGoogle Scholar