On the Olson and the Strong Davenport constants

Oscar Ordaz[1]; Andreas Philipp[2]; Irene Santos[1]; Wolfgang A. Schmid[3]

  • [1] Departamento de Matemáticas y Centro ISYS Facultad de Ciencias, Universidad Central de Venezuela Ap. 47567, Caracas 1041-A, Venezuela
  • [2] Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Graz Heinrichstraße 36 8010 Graz, Austria
  • [3] CMLS, École polytechnique 91128 Palaiseau cedex, France

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 3, page 715-750
  • ISSN: 1246-7405

Abstract

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A subset S of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of S is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, p -groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary p -groups of rank at most 2 , paralleling and building on recent results on this problem for the Olson constant.

How to cite

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Ordaz, Oscar, et al. "On the Olson and the Strong Davenport constants." Journal de Théorie des Nombres de Bordeaux 23.3 (2011): 715-750. <http://eudml.org/doc/219794>.

@article{Ordaz2011,
abstract = {A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary $p$-groups of rank at most $2$, paralleling and building on recent results on this problem for the Olson constant.},
affiliation = {Departamento de Matemáticas y Centro ISYS Facultad de Ciencias, Universidad Central de Venezuela Ap. 47567, Caracas 1041-A, Venezuela; Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Graz Heinrichstraße 36 8010 Graz, Austria; Departamento de Matemáticas y Centro ISYS Facultad de Ciencias, Universidad Central de Venezuela Ap. 47567, Caracas 1041-A, Venezuela; CMLS, École polytechnique 91128 Palaiseau cedex, France},
author = {Ordaz, Oscar, Philipp, Andreas, Santos, Irene, Schmid, Wolfgang A.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Davenport constant; Strong Davenport constant; Olson constant; zero-sumfree; zero-sum problem; zero sums; zero sum free sets; Davenport's constant; Olson's constant},
language = {eng},
month = {11},
number = {3},
pages = {715-750},
publisher = {Société Arithmétique de Bordeaux},
title = {On the Olson and the Strong Davenport constants},
url = {http://eudml.org/doc/219794},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Ordaz, Oscar
AU - Philipp, Andreas
AU - Santos, Irene
AU - Schmid, Wolfgang A.
TI - On the Olson and the Strong Davenport constants
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/11//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 3
SP - 715
EP - 750
AB - A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary $p$-groups of rank at most $2$, paralleling and building on recent results on this problem for the Olson constant.
LA - eng
KW - Davenport constant; Strong Davenport constant; Olson constant; zero-sumfree; zero-sum problem; zero sums; zero sum free sets; Davenport's constant; Olson's constant
UR - http://eudml.org/doc/219794
ER -

References

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  1. P. Baginski, The strong Davenport constant and the Olson constant. Unpublished manuscript, 2005. 
  2. G. Bhowmik and J.-Ch. Schlage-Puchta, An improvement on Olson’s constant for ℤ p ⊕ ℤ p . Acta Arith. 141 (2001), 311–319. Zbl1213.11041MR2587290
  3. G. Bhowmik and J.-Ch. Schlage-Puchta, Additive combinatorics and geometry of numbers. Manuscript. 
  4. É. Balandraud, An addition theorem and maximal zero-sumfree sets in ℤ / p ℤ . Israel J. Math, to appear. Zbl1287.11013
  5. S. T. Chapman, M. Freeze, and W. W. Smith, Minimal zero-sequences and the strong Davenport constant. Discrete Math. 203 (1999), 271–277. Zbl0944.20013MR1696250
  6. Ch. Delorme, A. Ortuño, and O. Ordaz, Some existence conditions for barycentric subsets. Rapport de Recherche N o 990, LRI, Paris-Sud, Orsay, France. 1995. 
  7. Ch. Delorme, I. Márquez, O. Ordaz, and A. Ortuño, Existence conditions for barycentric sequences. Discrete Math. 281 (2004), 163–172. Zbl1080.20020MR2047764
  8. J.-M. Deshouillers and G. Prakash, Large zero-free subsets of ℤ / p ℤ . Manuscript. 
  9. P. Erdős and H. Heilbronn, On the addition of residue classes mod p . Acta Arith. 9 (1964), 149–159. Zbl0156.04801MR166186
  10. M. Freeze, W. D. Gao, and A. Geroldinger, The critical number of finite abelain groups. J. Number Theory 129 (2009), 2766–2777. Zbl1214.11113MR2549531
  11. W. D. Gao and A. Geroldinger, On long minimal zero sequences in finite abelian groups. Period. Math. Hungar 38 (1999), 179–211. Zbl0980.11014MR1756238
  12. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24 (2006), 337–369. Zbl1122.11013MR2313123
  13. W. D. Gao, A. Geroldinger, and D. J. Grynkiewicz, Inverse zero-sum problems III. Acta Arith. 141 (2010), 103–152. Zbl1213.11178MR2579841
  14. W. D. Gao, I. Z. Ruzsa, and R. Thangadurai, Olson’s constant for the group ℤ p ⊕ ℤ p . J. Combin. Theory Ser. A 107 (2004), 49–67. Zbl1107.11014MR2063953
  15. A. Geroldinger, Additive group theory and non-unique factorizations. In: Combinatorial Number Theory and Additive Group Theory, pages 1–86, Advanced Course Math. CRM Barcelona, Birkhäuser Verlag, Basel, 2009. Zbl1221.20045MR2522037
  16. A. Geroldinger, D. Grynkiewicz, and W. A. Schmid, The catenary degree of Krull monoids I. J. Théor. Nombres Bordeaux 23 (2011), 137–169. Zbl1253.11101MR2780623
  17. A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory. Pure and Applied Mathematics, vol. 278, Chapman & Hall/CRC, 2006. Zbl1113.11002MR2194494
  18. A. Geroldinger, M. Liebmann, and A. Philipp, On the Davenport constant and on the structure of extremal zero-sum free sequences. Period. Math. Hung., to appear. Zbl1263.11038
  19. R. K. Guy, Unsolved problems in number theory, third edition. Problem Books in Mathematics, Springer-Verlag, New York, 2004. Zbl0474.10001MR2076335
  20. Y. O. Hamidoune and G. Zémor, On zero-free subset sums. Acta Arith. 78 (1996), 143–152. Zbl0863.11016MR1424536
  21. H. H. Nguyen, E. Szemerédi, and V. H. Vu, Subset sums modulo a prime. Acta Arith. 131 (2008), 303–316. Zbl1136.11016MR2383688
  22. H. H. Nguyen and V. H. Vu, Classification theorems for sumsets modulo a prime. J. Combin. Theory Ser. A 116 (2009), 936–959. Zbl1196.11048MR2513643
  23. H. H. Nguyen and V. H. Vu, An asymptotic characterization for incomplete sets in vector spaces. Submitted. 
  24. J. E. Olson, An addition theorem modulo p . J. Combin. Theory 5 (1968), 45–52. Zbl0174.05202MR227129
  25. J. E. Olson, Sums of sets of group elements. Acta Arith. 28 (1975/76), 147–156. Zbl0318.10035MR382215
  26. O. Ordaz and D. Quiroz, On zero free sets. Divulg. Mat. 14 (2006), 1–10. Zbl1217.20033MR2586356
  27. Ch. Reiher, A proof of the theorem according to which every prime number possesses Property B. Submitted. 
  28. S. Savchev and F. Chen, Long zero-free sequemces in finite cyclic groups. Discrete Math. 307 (2007), 2671–2679. Zbl1148.20038MR2351696
  29. W. A. Schmid, Inverse zero-sum problems II. Acta Arith. 143 (2010), 333–343. Zbl1219.11151MR2652584
  30. J. C. Subocz G., Some values of Olson’s constant. Divulg. Mat. 8 (2000), 121–128. Zbl0973.11025MR1802302
  31. E. Szemerédi, On a conjecture of Erdős and Heilbronn. Acta Arith. 17 (1970), 227–229. Zbl0222.10055MR268159
  32. P. Yuan, On the index of minimal zero-sum sequences over finte cyclic groups. J. Combin. Theory Ser. A 114 (2007), 1545–1551. Zbl1128.11011MR2360686

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