For a finite abelian group the Harborth constant is the smallest integer such that each squarefree sequence over of length , equivalently each subset of of cardinality at least , has a subsequence of length whose sum is . In this paper, it is established that for prime and .
Soit un groupe abélien fini. La constante de Harborth de , notée , est le plus petit entier tel que toute suite d’éléments deux à deux distincts de de longueur , de manière équivalente tout sous-ensemble de de cardinal au moins , admet une sous-suite de longueur dont la somme soit . Dans cet article, il est démontré que pour tout nombre premier et que .
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Keywords: finite abelian group, zero-sum problem, Harborth constant, squarefree sequence
@article{JTNB_2019__31_3_613_0, 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}, number = {3}, year = {2019}, doi = {10.5802/jtnb.1097}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jtnb.1097/} }
TY - JOUR AU - Guillot, Philippe AU - Marchan, Luz E. AU - Ordaz, Oscar AU - Schmid, Wolfgang A. AU - Zerdoum, Hanane TI - On the Harborth constant of $C_3 \oplus C_{3p}$ JO - Journal de théorie des nombres de Bordeaux PY - 2019 SP - 613 EP - 633 VL - 31 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.1097/ DO - 10.5802/jtnb.1097 LA - en ID - JTNB_2019__31_3_613_0 ER -
%0 Journal Article %A Guillot, Philippe %A Marchan, Luz E. %A Ordaz, Oscar %A Schmid, Wolfgang A. %A Zerdoum, Hanane %T On the Harborth constant of $C_3 \oplus C_{3p}$ %J Journal de théorie des nombres de Bordeaux %D 2019 %P 613-633 %V 31 %N 3 %I Société Arithmétique de Bordeaux %U http://www.numdam.org/articles/10.5802/jtnb.1097/ %R 10.5802/jtnb.1097 %G en %F JTNB_2019__31_3_613_0
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://www.numdam.org/articles/10.5802/jtnb.1097/
[1] Additive Combinatorics, A Menu of Research Problems, Discrete Mathematics and its Applications, CRC Press, 2018 | Zbl
[2] Progression-free sets in are exponentially small, Ann. Math., Volume 185 (2017) no. 1, pp. 331-337 | DOI | Zbl
[3] 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] On large subsets of with no three-term arithmetic progression, Ann. Math., Volume 185 (2017) no. 1, pp. 339-343 | DOI | Zbl
[5] A theorem in additive number theory, Bull. Res. Council Israel, Volume 10F (1961), pp. 41-43 | MR | Zbl
[6] Zero-sum problems in finite abelian groups: a survey, Expo. Math., Volume 24 (2006) no. 4, pp. 337-369 | MR | Zbl
[7] Inverse zero-sum problems, Acta Arith., Volume 128 (2007) no. 3, pp. 245-279 | MR | Zbl
[8] A variant of Kemnitz conjecture, J. Comb. Theory, Ser. A, Volume 107 (2004) no. 1, pp. 69-86 | MR | Zbl
[9] 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] Structural Additive Theory, Developments in Mathematics, 30, Springer, 2013 | MR | Zbl
[11] Ein Extremalproblem für Gitterpunkte, J. Reine Angew. Math., Volume 262-263 (1973), pp. 356-360 | MR | Zbl
[12] On a lattice point problem, Ars Comb., Volume 16-B (1983), pp. 151-160 | MR | Zbl
[13] Examining the maximum size of zero--sum-free subsets, Research Papers in Mathematics, Volume 19, Gettysburg College, 2016
[14] 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] Maximal caps in , Des. Codes Cryptography, Volume 46 (2008) no. 3, pp. 243-259 | DOI | MR | Zbl
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