Representation of finite abelian group elements by subsequence sums
Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 559-587.

Soit GC n 1 ...C n r un groupe abélien fini non trivial avec n 1 |n 2 |...|n r . Une conjecture d’Hamidoune dit que si W=w 1 ·...·w n est une suite d’entiers, tous, sauf au plus un, premiers à |G|, et S une suite d’éléments de G avec |S||W|+|G|-1|G|+1, la multiplicité maximale de S au plus |W|, et σ(W)0mod|G|, alors il existe un sous-groupe non trivial H tel que tout élément gH peut être représenté par une somme pondérée de la forme g= n i=1w i s i , avec s 1 ·...·s n une sous-suite de S. Nous donnons deux exemples qui montrent que cela n’est pas vrai en général, et nous caractérisons les contre-exemples pour les grands |W|1 2|G|.

Un théorème de Gao, généralisant un résultat plus ancien d’Olson, dit que si G est un groupe abélien fini, et S une suite d’éléments de G avec |S||G|+𝔻(G)-1, alors, soit tout élément de G peut être représenté par une sous-somme de S à |G| termes, soit il existe une classe g+H telle que tous sauf au plus |G/H|-2 termes de S sont dans g+H. Nous établissons quelques cas très spéciaux d’un analogue pondéré de ce théorème, conjecturé par Ordaz et Quiroz, et quelques conclusions partielles dans les autres cas, qui impliquent un résultat récent d’Ordaz et Quiroz. Cela est fait, en partie, en étendant un théorème de Grynkiewicz sur les partitions pondérées, que nous utilisons également pour améliorer le résultat de Gao cité précédemment en montrant que l’hypothèse |S||G|+𝔻(G)-1 peut être affaiblie en |S||G|+d * (G), où d * (G)= r i=1(n i -1). Nous utilisons aussi cette méthode pour déduire une variante de la conjecture d’Hamidoune valide si au moins d * (G) des w i sont premiers à |G|.

Let GC n 1 ...C n r be a finite and nontrivial abelian group with n 1 |n 2 |...|n r . A conjecture of Hamidoune says that if W=w 1 ·...·w n is a sequence of integers, all but at most one relatively prime to |G|, and S is a sequence over G with |S||W|+|G|-1|G|+1, the maximum multiplicity of S at most |W|, and σ(W)0mod|G|, then there exists a nontrivial subgroup H such that every element gH can be represented as a weighted subsequence sum of the form g= n i=1w i s i , with s 1 ·...·s n a subsequence of S. We give two examples showing this does not hold in general, and characterize the counterexamples for large |W|1 2|G|.

A theorem of Gao, generalizing an older result of Olson, says that if G is a finite abelian group, and S is a sequence over G with |S||G|+𝔻(G)-1, then either every element of G can be represented as a |G|-term subsequence sum from S, or there exists a coset g+H such that all but at most |G/H|-2 terms of S are from g+H. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis |S||G|+𝔻(G)-1 can be relaxed to |S||G|+d * (G), where d * (G)= r i=1(n i -1). We also use this method to derive a variation on Hamidoune’s conjecture valid when at least d * (G) of the w i are relatively prime to |G|.

DOI : 10.5802/jtnb.689
Classification : 11B75, 20K01
Mots clés : zero-sum problem, Davenport constant, weighted subsequence sums, setpartition, $\mathsf {d}^*(G)$
Grynkiewicz, David J. 1 ; Marchan, Luz E. 2 ; Ordaz, Oscar 3

1 Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Graz Heinrichstraße 36 8010 Graz, Austria.
2 Departamento de Matemáticas Decanato de Ciencias y Tecnologías Universidad Centroccidental Lisandro Alvarado Barquisimeto, Venezuela.
3 Departamento de Matemáticas y Centro ISYS Facultad de Ciencias Universidad Central de Venezuela Ap. 47567 Caracas 1041-A, Venezuela.
@article{JTNB_2009__21_3_559_0,
     author = {Grynkiewicz, David J. and Marchan, Luz E. and Ordaz, Oscar},
     title = {Representation of finite abelian group elements by subsequence sums},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {559--587},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     doi = {10.5802/jtnb.689},
     zbl = {1214.11034},
     mrnumber = {2605534},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.689/}
}
TY  - JOUR
AU  - Grynkiewicz, David J.
AU  - Marchan, Luz E.
AU  - Ordaz, Oscar
TI  - Representation of finite abelian group elements by subsequence sums
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2009
SP  - 559
EP  - 587
VL  - 21
IS  - 3
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.689/
DO  - 10.5802/jtnb.689
LA  - en
ID  - JTNB_2009__21_3_559_0
ER  - 
%0 Journal Article
%A Grynkiewicz, David J.
%A Marchan, Luz E.
%A Ordaz, Oscar
%T Representation of finite abelian group elements by subsequence sums
%J Journal de théorie des nombres de Bordeaux
%D 2009
%P 559-587
%V 21
%N 3
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.689/
%R 10.5802/jtnb.689
%G en
%F JTNB_2009__21_3_559_0
Grynkiewicz, David J.; Marchan, Luz E.; Ordaz, Oscar. Representation of finite abelian group elements by subsequence sums. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 559-587. doi : 10.5802/jtnb.689. http://archive.numdam.org/articles/10.5802/jtnb.689/

[1] Sukumar das Adhikari and Purusottam Rath, Davenport constant with weights and some related questions. Integers 6 (2006), A30, 6 pp (electronic). | MR | Zbl

[2] Sukumar das Adhikari and Yong-Gao Chen, Davenport constant with weights and some related questions II. J. Combin. Theory Ser. A 115 (2008), no. 1, 178–184. | MR

[3] N. Alon, A. Bialostocki and Y. Caro, The extremal cases in the Erdős-Ginzburg-Ziv Theorem. Unpublished.

[4] A. Bialostocki, P. Dierker, D. J. Grynkiewicz, and M. Lotspeich, On Some Developments of the Erdős-Ginzburg-Ziv Theorem II. Acta Arith. 110 (2003), no. 2, 173–184. | MR | Zbl

[5] Y. Caro, Zero-sum problems—a survey. Discrete Math. 152 (1996), no. 1–3, 93–113. | MR | Zbl

[6] P. Erdős, A. Ginzburg and A. Ziv, Theorem in Additive Number Theory. Bull. Res. Council Israel 10F (1961), 41–43. | Zbl

[7] W. Gao, Addition theorems for finite abelian groups. J. Number Theory 53 (1995), 241–246. | MR | Zbl

[8] W. Gao and A. Geroldinger, On Long Minimal Zero Sequences in Finite Abelian Groups. Periodica Math. Hungar. 38 (1999), no. 3, 179–211. | MR | Zbl

[9] W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: A survey. Expositiones Mathematicae, 24 (2006), no. 4, 337–369. | MR | Zbl

[10] W. Gao and W. Jin, Weighted sums in finite cyclic groups. Discrete Math. 283 (2004), no. 1-3, 243–247. | MR | Zbl

[11] A. Geroldinger and F. Halter-Koch, Non-unique factorizations: Algebraic, combinatorial and analytic theory. Pure and Applied Mathematics (Boca Raton) 278. Chapman & Hall/CRC, Boca Raton, FL, 2006. | MR | Zbl

[12] A. Geroldinger and R. Schneider, On Davenport’s Constant. J. Combin. Theory, Ser. A 61 (1992), no. 1, 147–152. | MR | Zbl

[13] S. Griffiths, The Erdős-Ginzberg-Ziv theorem with units. To appear in Discrete math. | MR

[14] D. J. Grynkiewicz, A Weighted Erdős-Ginzburg-Ziv Theorem. Combinatorica 26 (2006), no. 4, 445–453. | MR | Zbl

[15] D. J. Grynkiewicz, Quasi-periodic Decompositions and the Kemperman Structure Theorem, European J. Combin. 26 (2005), no. 5, 559–575. | MR | Zbl

[16] D. J. Grynkiewicz, On a Partition Analog of the Cauchy-Davenport Theorem. Acta Math. Hungar. 107 (2005), no. 1–2, 161–174. | MR | Zbl

[17] D. J. Grynkiewicz, On a conjecture of Hamidoune for subsequence sum., Integers 5 (2005), no. 2, A7, 11 pp. (electronic). | MR | Zbl

[18] D. J. Grynkiewicz and R. Sabar, Monochromatic and zero-sum sets of nondecreasing modified diameter. Electron. J. Combin. 13 (2006), no. 1, Research Paper 28, 19 pp. (electronic). | MR | Zbl

[19] D. J. Grynkiewicz, Sumsets, Zero-sums and Extremal Combinatorics. Ph. D. Dissertation, Caltech (2005).

[20] D. J. Grynkiewicz, A Step Beyond Kemperman’s Stucture Theorem. Preprint (2007). | MR

[21] Y. O. Hamidoune and A. Plagne, A new critical pair theorem applied to sum-free sets in abelian groups. Comment. Math. Helv. 79 (2004), no. 1, 183–207. | MR | Zbl

[22] Y. O. Hamidoune, On weighted sequence sums. Comb. Prob. Comput. 4 (1995), 363–367. | MR | Zbl

[23] Y. O. Hamidoune, On weighted sums in abelian groups. Discrete Math. 162 (1996), 127–132. | MR | Zbl

[24] T. Hungerford, Algebra. Springer-Verlag, New York, 1974. | MR | Zbl

[25] J. H. B. Kemperman, On Small Sumsets in an Abelian Group. Acta Math. 103 (1960), 63–88. | MR | Zbl

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

[27] M. Kneser, Ein Satz über abelsche Gruppen mit Anwendungen auf die Geometrie der Zahlen. Math. Z. 64 (1955), 429–434. | MR | Zbl

[28] S. Lang, Algebra. Third edition, Yale University, New Haven, CT, 1993.

[29] V. Lev, Critical pairs in abelian groups and Kemperman’s structure theorem. Int. J. Number Theory 2 (2006), no. 3, 379–396. | MR | Zbl

[30] M. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics 165, Springer-Verlag, New York, 1996. | MR | Zbl

[31] J. E. Olson, An addition theorem for finite abelian groups. J. Number Theory 9 (1977), no. 1, 63–70. | MR | Zbl

[32] O. Ordaz and D. Quiroz, Representation of group elements as subsequences sums. Discrete Mathematics 308 (2008), no. 15, 3315–3321. | MR | Zbl

[33] T. Tao and V. Vu, Additive Combinatorics. Cambridge Studies in Advanced Mathematics 105, Cambridge University Press, Cambridge, 2006. | MR | Zbl

Cité par Sources :