Restricted set addition in Abelian groups: results and conjectures

Vsevolod F. Lev[1]

  • [1] Department of Mathematics The University of Haifa at Oranim Tivon 36006, Israel

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 181-193
  • ISSN: 1246-7405

Abstract

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We present a system of interrelated conjectures which can be considered as restricted addition counterparts of classical theorems due to Kneser, Kemperman, and Scherk. Connections with the theorem of Cauchy-Davenport, conjecture of Erdős-Heilbronn, and polynomial method of Alon-Nathanson-Ruzsa are discussed.The paper assumes no expertise from the reader and can serve as an introduction to the subject.

How to cite

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Lev, Vsevolod F.. "Restricted set addition in Abelian groups: results and conjectures." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 181-193. <http://eudml.org/doc/249425>.

@article{Lev2005,
abstract = {We present a system of interrelated conjectures which can be considered as restricted addition counterparts of classical theorems due to Kneser, Kemperman, and Scherk. Connections with the theorem of Cauchy-Davenport, conjecture of Erdős-Heilbronn, and polynomial method of Alon-Nathanson-Ruzsa are discussed.The paper assumes no expertise from the reader and can serve as an introduction to the subject.},
affiliation = {Department of Mathematics The University of Haifa at Oranim Tivon 36006, Israel},
author = {Lev, Vsevolod F.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Restricted set addition; Kneser theorem; Kemperman-Scherk theorem},
language = {eng},
number = {1},
pages = {181-193},
publisher = {Université Bordeaux 1},
title = {Restricted set addition in Abelian groups: results and conjectures},
url = {http://eudml.org/doc/249425},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Lev, Vsevolod F.
TI - Restricted set addition in Abelian groups: results and conjectures
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 181
EP - 193
AB - We present a system of interrelated conjectures which can be considered as restricted addition counterparts of classical theorems due to Kneser, Kemperman, and Scherk. Connections with the theorem of Cauchy-Davenport, conjecture of Erdős-Heilbronn, and polynomial method of Alon-Nathanson-Ruzsa are discussed.The paper assumes no expertise from the reader and can serve as an introduction to the subject.
LA - eng
KW - Restricted set addition; Kneser theorem; Kemperman-Scherk theorem
UR - http://eudml.org/doc/249425
ER -

References

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  1. N. Alon, Combinatorial Nullstellensatz. Recent trends in combinatorics (Mátraháza, 1995). Combin. Probab. Comput. 8 (1–2) (1999), 7–29. Zbl0920.05026MR1684621
  2. N. Alon, M.B. Nathanson, I.Z. Ruzsa, Adding distinct congruence classes modulo a prime. American Math. Monthly 102 (1995), 250–255. Zbl0849.11081MR1317846
  3. N. Alon, M.B. Nathanson, I.Z. Ruzsa, The polynomial method and resricted sums of congruence classes. J. Number theory 56 (1996), 404–417. Zbl0861.11006MR1373563
  4. A. Cauchy, Recherches sur les nombres. Jour. Ecole polytechn. 9 (1813), 99–116. 
  5. H. Davenport, On the addition of residue classes. J. London Math. Soc. 10 (1935), 30–32. Zbl0010.38905
  6. —, A historical note. J. London Math. Soc. 22 (1947), 100–101. Zbl0029.34401MR22865
  7. J.A. Dias da Silva, Y.O. Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc. 26 (1994), 140–146. Zbl0819.11007MR1272299
  8. P. Erdős, R. Graham, Old and new problems and results in combinatorial number theory. L’Enseignement Mathématique, Geneva (1980). Zbl0434.10001MR592420
  9. G. Freiman, L. Low, J. Pitman, Sumsets with distinct summands and the conjecture of Erdős-Heilbronn on sums of residues. Astérisque 258 (1999), 163–172. Zbl0948.11008MR1701194
  10. J.H.B. Kemperman, On complexes in a semiroup. Indag. Math. 18 (1956), 247–254. Zbl0072.25605MR79005
  11. —, On small sumsets in an abelian group. Acta Math. 103 (1960), 63–88. Zbl0108.25704MR110747
  12. M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen. Math. Z. 58 (1953), 459–484. Zbl0051.28104MR56632
  13. —, Ein Satz über abelsche Gruppen mit Anwendungen auf die Geometrie der Zahlen. Math. Z. 61 (1955), 429–434. Zbl0064.04305MR68536
  14. V.F. Lev, Restricted set addition in groups, I. The classical setting. J.London Math. Soc. (2) 62 (2000), 27–40. Zbl0964.11016MR1771848
  15. —, Restricted set addition in groups, II. A generalization of the Erdős-Heilbronn conjecture. Electron. J. Combin. 7 (1) (2000), Research Paper 4, 10 pp. (electronic). Zbl0973.11026MR1742615
  16. —, Restricted set addition in groups, III. Integer sumsets with generic restrictions. Periodica Math. Hungarica 42 (2001), 89–98. Zbl1012.11020MR1832697
  17. H. B. Mann, Addition Theorems: The Addition Theorems of Group Theory and Number Theory. Interscience Publishers, a division of John Wiley and Sons, New York, 1965. Zbl0127.27203MR181626
  18. L.  Moser, Problem 4466. American Math. Monthly 58 (10) (1951), 703. 
  19. P. Scherk, Distinct elements in a set of sums (solution to Problem 4466). American Math. Monthly 62 (1) (1955), 46–47. 

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