An asymptotically tight bound for the Davenport constant

Benjamin Girard

doi:10.5802/jep.79

An asymptotically tight bound for the Davenport constant

Benjamin Girard ¹

¹ Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu - Paris Rive Gauche, IMJ-PRG F-75005, Paris, France

Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 605-611.

Abstract
Résumé

We prove that for every integer $r \geq 1$ the Davenport constant $D (C_{n}^{r})$ is asymptotic to $r n$ when $n$ tends to infinity. An extension of this theorem is also provided.

Nous prouvons que pour tout entier $r \geq 1$ , la constante de Davenport $D (C_{n}^{r})$ est équivalente à $r n$ lorsque $n$ tend vers l’infini. Nous proposons aussi une extension de ce théorème.

Received: 2018-02-19
Accepted: 2018-06-24
Published online: 2018-07-01

MR: 3852262 | Zbl: 1401.05311

DOI: 10.5802/jep.79

Classification: 05E15, 11B30, 11B75, 11A25, 20D60, 20K01
Keywords: Additive combinatorics, zero-sum sequences, Davenport constant, finite Abelian groups
Mot clés : Combinatoire additive, suites de somme nulle, constante de Davenport, groupes abéliens finis

Author's affiliations:

Benjamin Girard ¹

¹ Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu - Paris Rive Gauche, IMJ-PRG F-75005, Paris, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {An asymptotically tight bound for {the~Davenport} constant},
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Benjamin Girard. An asymptotically tight bound for the Davenport constant. Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 605-611. doi : 10.5802/jep.79. https://jep.centre-mersenne.org/articles/10.5802/jep.79/

References
Cited by

[1] W. R. Alford, A. Granville & C. Pomerance - “There are infinitely many Carmichael numbers”, Ann. of Math. (2) 139 (1994) no. 3, p. 703-722 | DOI | MR | Zbl

[2] N. Alon & M. Dubiner - “A lattice point problem and additive number theory”, Combinatorica 15 (1995) no. 3, p. 301-309 | DOI | MR | Zbl

[3] N. Alon, S. Friedland & G. Kalai - “Regular subgraphs of almost regular graphs”, J. Combin. Theory Ser. B 37 (1984) no. 1, p. 79-91 | DOI | MR | Zbl

[4] K. Cziszter, M. Domokos & A. Geroldinger - “The interplay of invariant theory with multiplicative ideal theory and with arithmetic combinatorics”, in Multiplicative ideal theory and factorization theory, Springer Proc. Math. Stat., vol. 170, Springer, 2016, p. 43-95 | DOI | MR | Zbl

[5] Y. Edel, C. Elsholtz, A. Geroldinger, S. Kubertin & L. Rackham - “Zero-sum problems in finite abelian groups and affine caps”, Q. J. Math. 58 (2007) no. 2, p. 159-186 | DOI | MR | Zbl

[6] Y. Edel, S. Ferret, I. Landjev & L. Storme - “The classification of the largest caps in $AG (5, 3)$ ”, J. Combin. Theory Ser. A 99 (2002) no. 1, p. 95-110 | DOI | MR | Zbl

[7] J. S. Ellenberg & D. Gijswijt - “On large subsets of $𝔽_{q}^{n}$ with no three-term arithmetic progression”, Ann. of Math. (2) 185 (2017) no. 1, p. 339-343 | DOI | MR | Zbl

[8] P. van Emde Boas - A combinatorial problem on finite abelian groups. II (1969) no. ZW-007, 60 pages, Technical report | MR | Zbl

[9] P. van Emde Boas & D. Kruyswijk - A combinatorial problem on finite abelian groups. III (1969) no. ZW-008, Technical report | Zbl

[10] W. Gao & A. Geroldinger - “Zero-sum problems and coverings by proper cosets”, European J. Combin. 24 (2003) no. 5, p. 531-549 | DOI | MR | Zbl

[11] W. Gao & A. Geroldinger - “Zero-sum problems in finite abelian groups: a survey”, Exposition. Math. 24 (2006) no. 4, p. 337-369 | DOI | MR | Zbl

[12] W. D. Gao, Q. H. Hou, W. A. Schmid & R. Thangadurai - “On short zero-sum subsequences. II”, Integers 7 (2007), article #A21 | MR | Zbl

[13] A. Geroldinger - “Additive group theory and non-unique factorizations”, in Combinatorial number theory and additive group theory, Adv. Courses Math. CRM Barcelona, Birkhäuser Verlag, Basel, 2009, p. 1-86 | Zbl

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

[15] A. Geroldinger, M. Liebmann & A. Philipp - “On the Davenport constant and on the structure of extremal zero-sum free sequences”, Period. Math. Hungar. 64 (2012) no. 2, p. 213-225 | DOI | MR | Zbl

[16] A. Geroldinger & R. Schneider - “On Davenport’s constant”, J. Combin. Theory Ser. A 61 (1992) no. 1, p. 147-152 | DOI | MR | Zbl

[17] H. Harborth - “Ein Extremalproblem für Gitterpunkte”, J. reine angew. Math. 262/263 (1973), p. 356-360 | MR | Zbl

[18] J. Kaczorowski - “On the distribution of irreducible algebraic integers”, Monatsh. Math. 156 (2009) no. 1, p. 47-71 | MR | Zbl

[19] M. Mazur - “A note on the growth of Davenport’s constant”, Manuscripta Math. 74 (1992) no. 3, p. 229-235 | DOI | MR | Zbl

[20] R. Meshulam - “An uncertainty inequality and zero subsums”, Discrete Math. 84 (1990) no. 2, p. 197-200 | MR | Zbl

[21] W. Narkiewicz - Elementary and analytic theory of algebraic numbers, Springer Monographs in Math., Springer-Verlag, Berlin, 2004 | DOI | Zbl

[22] J. E. Olson - “A combinatorial problem on finite Abelian groups. I”, J. Number Theory 1 (1969), p. 8-10 | DOI | MR | Zbl

[23] J. E. Olson - “A combinatorial problem on finite Abelian groups. II”, J. Number Theory 1 (1969), p. 195-199 | DOI | MR | Zbl

[24] G. Pellegrino - “The maximal order of the spherical cap in $S_{4, 3}$ ”, Matematiche 25 (1971), p. 149-157

[25] A. Potechin - “Maximal caps in $AG (6, 3)$ ”, Des. Codes Cryptogr. 46 (2008) no. 3, p. 243-259 | DOI | MR | Zbl

[26] K. Rogers - “A combinatorial problem in Abelian groups”, Math. Proc. Cambridge Philos. Soc. 59 (1963), p. 559-562 | DOI | MR | Zbl

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