Abstract
Let G be a finite abelian group and \(A\subset \mathbb Z\). The weighted zero-sum constant \(s_A(G)\) (resp. \(\eta _A(G)\)) is defined as the least positive integer t, such that every sequence S over G with length \(\ge t\) has an A-weighted zero-sum subsequence of length \(\mathrm{exp}(G)\) (resp. \(\le \!\!\exp (G)\)). In this article, we investigate the value of \(s_A(G)\) and \(\eta _A(G)\) in the case \(G=\mathbb Z_n\oplus \mathbb Z_n\oplus \cdots \oplus \mathbb Z_n\), where n is a square-free odd integer and A is the set of integers co-prime to n. We also obtain certain properties about extremal zero-sum free sequences.
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References
Adhikari S D, Ambily A A and Sury B, Zero-sum problems with subgroup weights, Proc. Indian Acad. Sci. 120(3) (2010) 259–266
Adhikari S D, Chen Y G, Friedlander J B, Konyagin S V and Pappalardi F, Contributions to zero-sum problems, Discrete Math. 306(1) (2006) 1–10
Chintamani M and Paul P, On Some weighted Zero-sum Constants, Int. J. Number Theory 13(2) (2017) 301–308
Chintamani M and Paul P, On some weighted zero-sum constants II, Int. J. Number Theory 14(2) (2018) 383–397
Erdős P, Ginzberg A and Ziv A, A Theorem in the additive number theory, Bull. Res. Council Israel 10F (1961) 41–43
Gao W D, Hou Q H, Schmid W A and Thangadurai R, On short zero-sum subsequences. II, Integers 7 (2007) A21
Godinho H, Lemos A and Marques D, Weighted zero-sum problems over \(C_r^3\), Algebra Discrete Math. 15 (2013) 201–212
Griffiths S, The Erdős–Ginzberg–Ziv theorem with units, Discrete Math. 308(23) (2008) 5473–5484
Luca F, A generalization of a classical zero-sum problem, Discrete Math. 307(13) (2007) 1672–1678
Oliveira F A A, Lemos A and Godinho H T, Weighted EGZ-constant for \(p\)-groups of rank 2, Int. J. Number Theory 15(10) (2019) 2163–2177
Paul P, Counting zero-free sequences, J. Comb. Number Theory 1(1) (2015) 65–73
Acknowledgements
A part of this work was carried out when the authors visited Harish-Chandra Research Institute, Prayagraj (Allahabad). The authors would like to thank Prof. R Thangadurai for the invitation and the hospitality. The authors are thankful to the anonymous referee for his/her valuable comments. The first-named author (MC) was supported by MATRICS, DST-SERB Grant (MTR/2019/000004) during the project
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Communicated by Sanoli Gun.
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Chintamani, M., Paul, P. A weighted Erdős–Ginzburg–Ziv constant for finite abelian groups with higher rank. Proc Math Sci 132, 13 (2022). https://doi.org/10.1007/s12044-022-00671-w
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DOI: https://doi.org/10.1007/s12044-022-00671-w