On the Harborth constant of C 3 C 3p
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 613-633.

For a finite abelian group (G,+,0) the Harborth constant g(G) is the smallest integer k such that each squarefree sequence over G of length k, equivalently each subset of G of cardinality at least k, has a subsequence of length exp(G) whose sum is 0. In this paper, it is established that g(C 3 C 3p )=3p+3 for prime p3 and g(C 3 C 9 )=13.

Soit (G,+,0) un groupe abélien fini. La constante de Harborth de G, notée g(G), est le plus petit entier k tel que toute suite d’éléments deux à deux distincts de G de longueur k, de manière équivalente tout sous-ensemble de G de cardinal au moins k, admet une sous-suite de longueur exp(G) dont la somme soit 0. Dans cet article, il est démontré que g(C 3 C 3p )=3p+3 pour tout nombre premier p3 et que g(C 3 C 9 )=13.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1097
Classification: 11B30,  20K01
Keywords: finite abelian group, zero-sum problem, Harborth constant, squarefree sequence
Guillot, Philippe 1, 2; Marchan, Luz E. 3; Ordaz, Oscar 4; Schmid, Wolfgang A. 1, 2; Zerdoum, Hanane 1, 2

1 Laboratoire Analyse, Géométrie et Applications, LAGA, Université Sorbonne Paris Nord, CNRS, UMR 7539, F-93430, Villetaneuse, France
2 Laboratoire Analyse, Géométrie et Applications (LAGA, UMR 7539), COMUE Université Paris Lumières, Université Paris 8, CNRS, 93526 Saint-Denis cedex, France
3 Escuela Superior Politécnica del Litoral, ESPOL, Facultad de ciencias naturales y matemática. Campus Gustavo Galindo, km 30.5, vía Perimetral, P.O. Box 09-01-5863, Guayaquil, Ecuador
4 Escuela de Matemáticas y Laboratorio MoST, Centro ISYS, Facultad de Ciencias, Universidad Central de Venezuela, Ap. 47567, Caracas 1041–A, Venezuela
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     title = {On the {Harborth} constant of $C_3 \oplus C_{3p}$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Guillot, Philippe; Marchan, Luz E.; Ordaz, Oscar; Schmid, Wolfgang A.; Zerdoum, Hanane. On the Harborth constant of $C_3 \oplus C_{3p}$. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 613-633. doi : 10.5802/jtnb.1097. http://archive.numdam.org/articles/10.5802/jtnb.1097/

[1] Bajnok, Béla Additive Combinatorics, A Menu of Research Problems, Discrete Mathematics and its Applications, CRC Press, 2018 | Zbl

[2] Croot, Ernie; Lev, Vsevolod F.; Pach, Péter Pál Progression-free sets in Z 4 n are exponentially small, Ann. Math., Volume 185 (2017) no. 1, pp. 331-337 | DOI | Zbl

[3] Edel, Yves; Elsholtz, Christian; Geroldinger, Alfred; Kubertin, Silke; Rackham, Laurence Zero-sum problems in finite abelian groups and affine caps, Q. J. Math, Volume 58 (2007) no. 2, pp. 159-186 | DOI | MR | Zbl

[4] Ellenberg, Jordan S.; Gijswijt, Dion On large subsets of 𝔽 q n with no three-term arithmetic progression, Ann. Math., Volume 185 (2017) no. 1, pp. 339-343 | DOI | Zbl

[5] Erdős, Paul; Ginzburg, A.; Ziv, A. A theorem in additive number theory, Bull. Res. Council Israel, Volume 10F (1961), pp. 41-43 | MR | Zbl

[6] Gao, Weidong D.; Geroldinger, Alfred Zero-sum problems in finite abelian groups: a survey, Expo. Math., Volume 24 (2006) no. 4, pp. 337-369 | MR | Zbl

[7] Gao, Weidong D.; Geroldinger, Alfred; Schmid, Wolfgang A. Inverse zero-sum problems, Acta Arith., Volume 128 (2007) no. 3, pp. 245-279 | MR | Zbl

[8] Gao, Weidong D.; Thangadurai, Ravindranathan A variant of Kemnitz conjecture, J. Comb. Theory, Ser. A, Volume 107 (2004) no. 1, pp. 69-86 | MR | Zbl

[9] Geroldinger, Alfred Additive group theory and non-unique factorizations, Combinatorial number theory and additive group theory (Advanced Courses in Mathematics - CRM Barcelona), Birkhäuser, 2009, pp. 1-86 | Zbl

[10] Grynkiewicz, David J. Structural Additive Theory, Developments in Mathematics, 30, Springer, 2013 | MR | Zbl

[11] Harborth, Heiko Ein Extremalproblem für Gitterpunkte, J. Reine Angew. Math., Volume 262-263 (1973), pp. 356-360 | MR | Zbl

[12] Kemnitz, Arnfried On a lattice point problem, Ars Comb., Volume 16-B (1983), pp. 151-160 | MR | Zbl

[13] Kiefer, C. Examining the maximum size of zero-h-sum-free subsets, Research Papers in Mathematics, Volume 19, Gettysburg College, 2016

[14] Marchan, Luz E.; Ordaz, Oscar; Ramos, Dennys; Schmid, Wolfgang A. Some exact values of the Harborth constant and its plus-minus weighted analogue, Arch. Math., Volume 101 (2013) no. 6, pp. 501-512 | DOI | MR | Zbl

[15] Potechin, Aaron Maximal caps in AG (6,3), Des. Codes Cryptography, Volume 46 (2008) no. 3, pp. 243-259 | DOI | MR | Zbl

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