On the inverse problems associated with subsequence sums of zero-sum free sequences over finite abelian groups Ⅱ

Rui Wang; Jiangtao Peng; Rui Wang; Jiangtao Peng

doi:10.3934/math.2021101

AIMS Mathematics

2021, Volume 6, Issue 2: 1706-1714. doi: 10.3934/math.2021101

Previous Article Next Article

Research article

On the inverse problems associated with subsequence sums of zero-sum free sequences over finite abelian groups Ⅱ

Rui Wang ,
Jiangtao Peng ^,

College of Science, Civil Aviation University of China, Tianjin, P. R. China

Received: 17 October 2020 Accepted: 20 November 2020 Published: 27 November 2020
MSC : 11P70, 11B75

Let $G$ be an additive finite abelian group with exponent $\exp(G)$ and $S$ be a sequence with elements of $G$. Let $\Sigma(S) \subset G$ denote the set of group elements which can be expressed as the sum of a nonempty subsequence of $S$. We say $S$ is zero-sum free if $0 \not\in \Sigma(S)$. In this paper, we determine the structures of the zero-sum free sequences $S$ such that $|S| = \exp(G)+2$ and $|\Sigma(S)| = 4\exp(G)-1$, which partly confirms a conjecture of J. Peng et al.
- abelian groups,
- Davenport constant,
- inverse problems,
- subsequence sums,
- zero-sum free sequences
Citation: Rui Wang, Jiangtao Peng. On the inverse problems associated with subsequence sums of zero-sum free sequences over finite abelian groups Ⅱ[J]. AIMS Mathematics, 2021, 6(2): 1706-1714. doi: 10.3934/math.2021101

Related Papers:

Abstract

Let $G$ be an additive finite abelian group with exponent $\exp(G)$ and $S$ be a sequence with elements of $G$. Let $\Sigma(S) \subset G$ denote the set of group elements which can be expressed as the sum of a nonempty subsequence of $S$. We say $S$ is zero-sum free if $0 \not\in \Sigma(S)$. In this paper, we determine the structures of the zero-sum free sequences $S$ such that $|S| = \exp(G)+2$ and $|\Sigma(S)| = 4\exp(G)-1$, which partly confirms a conjecture of J. Peng et al.

References

[1]	B. Bollobás, I. Leader, The number of k-sums modulo k, J. Number Theory, 78 (1999), 27-35. doi: 10.1006/jnth.1999.2405
[2]	R. B. Eggletőn, P. Erdos, Two combinatorial problems in group theory, Acta Arith., 21 (1972), 111-116. doi: 10.4064/aa-21-1-111-116
[3]	W. Gao, M. Huang, W. Hui, Y. Li, C. Liu, J. Peng, Sums of sets of abelian group elements, J. Number Theory, 208 (2020), 208-229. doi: 10.1016/j.jnt.2019.07.026
[4]	W. Gao, I. Leader, Sums and k-sums in an abelian groups of order k, J. Number Theory, 120 (2006), 26-32. doi: 10.1016/j.jnt.2005.11.010
[5]	W. Gao, Y. Li, J. Peng, F. Sun, Subsums of a Zero-sum Free Subset of an Abelian Group, Electron. J. Combin., 15 (2008), R116. doi: 10.37236/840
[6]	W. Gao, Y. Li, J. Peng, F. Sun, On subsequence sums of a zero-sum free sequence Ⅱ, Electron. J. Combin., 15 (2008), R117. doi: 10.37236/841
[7]	A. Geroldinger, D. J. Grynkiewicz, The large Davenport constant I: Groups with a cyclic, index 2 subgroup, J. Pure Appl. Algebra, 217 (2013), 863-885. doi: 10.1016/j.jpaa.2012.09.004
[8]	A. Geroldinger, F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, Chapman & Hall/CRC, 2006, vol. 278, p700.
[9]	H. Guan, X. Zeng, P. Yuan, Description of invariant F(5) of a zero-sum free sequence, Acta Sci. Natur. Univ. Sunyatseni, 49 (2010), 1-4 (In Chinese).
[10]	J. E. Olson, Sums of sets of group elements, Acta Arith., 28 (1975), 147-156. doi: 10.4064/aa-28-2-147-156
[11]	J. E. Olson, E. T. White, Sums from a sequence of group elements, Number Theory and Algebra (H. Zassenhaus, ed.), Academic Press, 1977,215-222.
[12]	J. Peng, W. Hui, On the structure of zero-sum free set with minimum subset sums in abelian groups, Ars Combin., 146 (2019), 63-74.
[13]	J. Peng, W. Hui, on subsequence sums of zero-sum free sequences in abelian groups of rank two. JP J. Algebra, Number Theory Appl., 48 (2020), 133-153.
[14]	J. Peng, W. Hui, Y. Li, F. Sun, On subset sums of zero-sum free sets of abelian groups. Int. J. Number Theory, 15 (2019), 645-654. doi: 10.1142/S1793042119500349
[15]	J. Peng, Y. Li, C. Liu, M. Huang, On the inverse problems associated with subsequence sums of zero-sum free sequences over finite abelian groups, Colloq. Math., 163 (2021), 317-332. doi: 10.4064/cm8033-12-2019
[16]	J. Peng, Y. Li, C. Liu, M. Huang, On subsequence sums of a zero-sum free sequence over finite abelian groups, J. Number Theory, 217 (2020), 193-217. doi: 10.1016/j.jnt.2020.04.024
[17]	J. Peng, Y. Qu, Y. Li, On the inverse problems associated with subsequence sums in C_p⊕C_p, Front. Math. China, 15 (2020), 985-1000. doi: 10.1007/s11464-020-0869-2
[18]	A. Pixton, Sequences with small subsums sets, J. Number Theory, 129 (2009), 806-817. doi: 10.1016/j.jnt.2008.12.004
[19]	Y. Qu, X. Xia, L. Xue, Q. Zhong, Subsequence sums of zero-sum free sequences over finite abelian groups, Colloq. Math., 140 (2015), 119-127. doi: 10.4064/cm140-1-10
[20]	F. Sun, On subsequence sums of a zero-sum free sequence, Electron. J. Combin., 14 (2007), R52. doi: 10.37236/970
[21]	F. Sun, Y. Li, J. Peng, A note on the inverse problems associated with subsequence sums, J. Combin. Math. Combin. Comput. (in press).
[22]	P. Yuan, Subsequence sums of a zero-sumfree sequence, European J. Combin., 30 (2009), 439-446. doi: 10.1016/j.ejc.2008.04.008
[23]	P. Yuan, Subsequence Sums of Zero-sum-free Sequences, Electron. J. Combin., 16 (2009), R97. doi: 10.37236/186
[24]	P. Yuan, X. Zeng, On zero-sum free subsets of length 7, Electron. J. Combin., 17 (2010), R104. doi: 10.37236/376

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)