Restricted set addition in Abelian groups: results and conjectures
Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 181-193.

Nous présentons un ensemble de conjectures imbriquées qui peuvent être considérées comme des analogues pour l’addition restreinte des théorèmes classiques dûs à Kneser, Kemperman et Scherk. Les connections avec le théorème de Cauchy-Davenport, la conjecture d’Erdős-Heilbronn et la méthode polynomiale d’Alon-Nathanson-Ruzsa sont étudiées.

Cet article ne suppose pas d’expertise de la part du lecteur et peut servir d’introduction au sujet.

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.

DOI : 10.5802/jtnb.485
Lev, Vsevolod F. 1

1 Department of Mathematics The University of Haifa at Oranim Tivon 36006, Israel
@article{JTNB_2005__17_1_181_0,
     author = {Lev, Vsevolod F.},
     title = {Restricted set addition in {Abelian} groups:  results and conjectures},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {181--193},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     doi = {10.5802/jtnb.485},
     zbl = {1162.11318},
     mrnumber = {2152219},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.485/}
}
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
SP  - 181
EP  - 193
VL  - 17
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.485/
DO  - 10.5802/jtnb.485
LA  - en
ID  - JTNB_2005__17_1_181_0
ER  - 
%0 Journal Article
%A Lev, Vsevolod F.
%T Restricted set addition in Abelian groups:  results and conjectures
%J Journal de théorie des nombres de Bordeaux
%D 2005
%P 181-193
%V 17
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.485/
%R 10.5802/jtnb.485
%G en
%F JTNB_2005__17_1_181_0
Lev, Vsevolod F. Restricted set addition in Abelian groups:  results and conjectures. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 181-193. doi : 10.5802/jtnb.485. http://www.numdam.org/articles/10.5802/jtnb.485/

[A99] N. Alon, Combinatorial Nullstellensatz. Recent trends in combinatorics (Mátraháza, 1995). Combin. Probab. Comput. 8 (1–2) (1999), 7–29. | MR | Zbl

[ANR95] N. Alon, M.B. Nathanson, I.Z. Ruzsa, Adding distinct congruence classes modulo a prime. American Math. Monthly 102 (1995), 250–255. | MR | Zbl

[ANR96] N. Alon, M.B. Nathanson, I.Z. Ruzsa, The polynomial method and resricted sums of congruence classes. J. Number theory 56 (1996), 404–417. | MR | Zbl

[C13] A. Cauchy, Recherches sur les nombres. Jour. Ecole polytechn. 9 (1813), 99–116.

[D35] H. Davenport, On the addition of residue classes. J. London Math. Soc. 10 (1935), 30–32. | JFM | Zbl

[D47] —, A historical note. J. London Math. Soc. 22 (1947), 100–101. | MR | Zbl

[DH94] J.A. Dias da Silva, Y.O. Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc. 26 (1994), 140–146. | MR | Zbl

[EG80] P. Erdős, R. Graham, Old and new problems and results in combinatorial number theory. L’Enseignement Mathématique, Geneva (1980). | MR | Zbl

[FLP99] 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. | Numdam | MR | Zbl

[Ke56] J.H.B. Kemperman, On complexes in a semiroup. Indag. Math. 18 (1956), 247–254. | MR | Zbl

[Ke60] —, On small sumsets in an abelian group. Acta Math. 103 (1960), 63–88. | MR | Zbl

[Kn53] M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen. Math. Z. 58 (1953), 459–484. | MR | Zbl

[Kn55] —, Ein Satz über abelsche Gruppen mit Anwendungen auf die Geometrie der Zahlen. Math. Z. 61 (1955), 429–434. | MR | Zbl

[L00a] V.F. Lev, Restricted set addition in groups, I. The classical setting. J.London Math. Soc. (2) 62 (2000), 27–40. | MR | Zbl

[L00b] —, 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). | MR | Zbl

[L01] —, Restricted set addition in groups, III. Integer sumsets with generic restrictions. Periodica Math. Hungarica 42 (2001), 89–98. | MR | Zbl

[Ma65] 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. | MR | Zbl

[Mo51] L.  Moser, Problem 4466. American Math. Monthly 58 (10) (1951), 703.

[S55] P. Scherk, Distinct elements in a set of sums (solution to Problem 4466). American Math. Monthly 62 (1) (1955), 46–47.

Cité par Sources :