On the Harborth constant of $C_3 \oplus C_{3p}$

Guillot, Philippe; Marchan, Luz E.; Ordaz, Oscar; Schmid, Wolfgang A.; Zerdoum, Hanane

doi:10.5802/jtnb.1097

On the Harborth constant of

C_{3} \oplus C_{3 p}

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

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 613-633.

Résumé
Abstract

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 $g (C_{3} \oplus C_{3 p}) = 3 p + 3$ pour tout nombre premier $p \neq 3$ et que $g (C_{3} \oplus C_{9}) = 13$ .

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 $g (C_{3} \oplus C_{3 p}) = 3 p + 3$ for prime $p \neq 3$ and $g (C_{3} \oplus C_{9}) = 13$ .

Reçu le : 2018-09-22
Révisé le : 2019-09-26
Accepté le : 2019-10-25
Publié le : 2020-05-06

DOI : 10.5802/jtnb.1097

Classification : 11B30, 20K01
Mots clés : finite abelian group, zero-sum problem, Harborth constant, squarefree sequence

Affiliations des auteurs :

Guillot, Philippe ^{1, 2} ; Marchan, Luz E. ³ ; Ordaz, Oscar ⁴ ; Schmid, Wolfgang A. ^{1, 2} ; Zerdoum, Hanane ^{1, 2}

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     author = {Guillot, Philippe and Marchan, Luz E. and Ordaz, Oscar and Schmid, Wolfgang A. and Zerdoum, Hanane},
     title = {On the {Harborth} constant of $C_3 \oplus C_{3p}$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {613--633},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
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     year = {2019},
     doi = {10.5802/jtnb.1097},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.1097/}
}

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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, Tome 31 (2019) no. 3, pp. 613-633. doi : 10.5802/jtnb.1097. http://archive.numdam.org/articles/10.5802/jtnb.1097/

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