scholarly journals On the Number of Subsequences with a Given Sum in a Finite Abelian Group

10.37236/620 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Gerard Jennhwa Chang ◽  
Sheng-Hua Chen ◽  
Yongke Qu ◽  
Guoqing Wang ◽  
Haiyan Zhang
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Zero Sum

Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for $N_g(S)$. In particular, we prove that either $N_g(S)=0$ or $N_g(S)\ge2^{|S|-D(G)+1}$, where $D(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nonempty zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.

On zero-sum subsequences of prescribed length

2017 ◽  
Vol 14 (01) ◽  
pp. 167-191 ◽  
Author(s):  
Dongchun Han ◽  
Hanbin Zhang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Davenport Constant ◽  
Zero Sum

Let [Formula: see text] be an additive finite abelian group with exponent [Formula: see text]. For any positive integer [Formula: see text], let [Formula: see text] be the smallest positive integer [Formula: see text] such that every sequence [Formula: see text] in [Formula: see text] of length at least [Formula: see text] has a zero-sum subsequence of length [Formula: see text]. Let [Formula: see text] be the Davenport constant of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite abelian [Formula: see text]-group with [Formula: see text] then [Formula: see text] for every [Formula: see text], which confirms a conjecture by Gao et al. recently, where [Formula: see text] is a prime.

On some weighted zero-sum constants

2017 ◽  
Vol 13 (02) ◽  
pp. 301-308 ◽  
Author(s):  
Mohan N. Chintamani ◽  
Prabal Paul
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Structural Information ◽  
Finite Abelian Group ◽  
Text Length ◽  
Zero Sum

Let [Formula: see text] be a finite abelian group with exponent exp[Formula: see text]. Let [Formula: see text]. The constant [Formula: see text] is defined as the least positive integer [Formula: see text] such that for any given sequence [Formula: see text] of elements of [Formula: see text] with length [Formula: see text] it has a [Formula: see text] length [Formula: see text]-weighted zero-sum subsequence. In this article, we obtain the exact value of [Formula: see text] for [Formula: see text] and an upper bound for the case [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] is an odd integer and [Formula: see text]. We also obtain the structural information on the extremal zero-sum free sequences.

scholarly journals Subsums of a Zero-sum Free Subset of an Abelian Group

10.37236/840 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Weidong Gao ◽  
Yuanlin Li ◽  
Jiangtao Peng ◽  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Cyclic Group ◽  
Open Problem ◽  
Nonempty Subset ◽  
Finite Abelian Group ◽  
Positive Answer ◽  
Free Subset ◽  
Zero Sum ◽  
G 28

Let $G$ be an additive finite abelian group and $S \subset G$ a subset. Let f$(S)$ denote the number of nonzero group elements which can be expressed as a sum of a nonempty subset of $S$. It is proved that if $|S|=6$ and there are no subsets of $S$ with sum zero, then f$(S)\geq 19$. Obviously, this lower bound is best possible, and thus this result gives a positive answer to an open problem proposed by R.B. Eggleton and P. Erdős in 1972. As a consequence, we prove that any zero-sum free sequence $S$ over a cyclic group $G$ of length $|S| \ge {6|G|+28\over19}$ contains some element with multiplicity at least ${6|S|-|G|+1\over17}$.

scholarly journals The zero-sum constant, the Davenport constant and their analogues

2020 ◽  
pp. 1-14
Author(s):  
Maciej Zakarczemny
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Asymptotic Behaviour ◽  
Finite Abelian Group ◽  
Classical Case ◽  
Independent Interest ◽  
Davenport Constant ◽  
Smooth Numbers ◽  
Abelian Case ◽  
Zero Sum

Let D(G) be the Davenport constant of a finite Abelian group G. For a positive integer m (the case m=1, is the classical case) let Em(G) (or ηm(G)) be the least positive integer t such that every sequence of length t in G contains m disjoint zero-sum sequences, each of length |G| (or of length ≤exp(G), respectively). In this paper, we prove that if G is an Abelian group, then Em(G)=D(G)–1+m|G|, which generalizes Gao’s relation. Moreover, we examine the asymptotic behaviour of the sequences (Em(G))m≥1 and (ηm(G))m≥1. We prove a generalization of Kemnitz’s conjecture. The paper also contains a result of independent interest, which is a stronger version of a result by Ch. Delorme, O. Ordaz, D. Quiroz. At the end, we apply the Davenport constant to smooth numbers and make a natural conjecture in the non-Abelian case.

On the Erdős–Ginzburg–Ziv invariant and zero-sum Ramsey number for intersecting families

2014 ◽  
Vol 10 (07) ◽  
pp. 1637-1647 ◽  
Author(s):  
Haiyan Zhang ◽  
Guoqing Wang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Zero Sum

Let G be a finite abelian group, and let m > 0 with exp (G) | m. Let sm(G) be the generalized Erdős–Ginzburg–Ziv invariant which denotes the smallest positive integer d such that any sequence of elements in G of length d contains a subsequence of length m with sum zero in G. For any integer r > 0, let [Formula: see text] be the collection of all r-uniform intersecting families of size m. Let [Formula: see text] be the smallest positive integer d such that any G-coloring of the edges of the complete r-uniform hypergraph [Formula: see text] yields a zero-sum copy of some intersecting family in [Formula: see text]. Among other results, we mainly prove that [Formula: see text], where Ω(sm(G)) denotes the least positive integer n such that [Formula: see text], and we show that if r | Ω(sm(G)) – 1 then [Formula: see text].

On some weighted zero-sum constants II

2018 ◽  
Vol 14 (02) ◽  
pp. 383-397 ◽  
Author(s):  
Mohan N. Chintamani ◽  
Prabal Paul
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Structure Theorem ◽  
Finite Abelian Group ◽  
Zero Sum

For a finite abelian group [Formula: see text] with exponent [Formula: see text], let [Formula: see text]. The constant [Formula: see text] (respectively [Formula: see text]) is defined to be the least positive integer [Formula: see text] such that given any sequence [Formula: see text] over [Formula: see text] with length [Formula: see text] has a [Formula: see text]-weighted zero-sum subsequence of length [Formula: see text] (respectively at most [Formula: see text]). In [M. N. Chintamani and P. Paul, On some weighted zero-sum constants, Int. J. Number Theory 13(2) (2017) 301–308], we proved the exact value of this constant for the group [Formula: see text] and proved the structure theorem for the extremal sequences related to this constant. In this paper, we prove the similar results for the group [Formula: see text] and we obtained an upper bound when [Formula: see text] is replaced by any integer [Formula: see text].

Zero-sum subsequences of distinct lengths

2015 ◽  
Vol 11 (07) ◽  
pp. 2141-2150 ◽  
Author(s):  
Weidong Gao ◽  
Pingping Zhao ◽  
Jujuan Zhuang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Zero Sum ◽  
Positive Integers

Let G be an additive finite abelian group, and let disc (G) denote the smallest positive integer t such that every sequence S over G of length ∣S∣ ≥ t has two nonempty zero-sum subsequences of distinct lengths. We determine disc (G) for some groups including the groups [Formula: see text], the groups of rank at most two and the groups Cmpn ⊕ H, where m, n are positive integers, p is a prime and H is a p-group with pn ≥ D*(H).

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group

2021 ◽  
Author(s):  
Weidong Gao ◽  
Siao Hong ◽  
Wanzhen Hui ◽  
Xue Li ◽  
Qiuyu Yin ◽  
...  
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum

scholarly journals On Subsequence Sums of a Zero-sum Free Sequence

10.37236/970 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum ◽  
Open Question

Let $G$ be a finite abelian group with exponent $m$, and let $S$ be a sequence of elements in $G$. Let $f(S)$ denote the number of elements in $G$ which can be expressed as the sum over a nonempty subsequence of $S$. In this paper, we show that, if $|S|=m$ and $S$ contains no nonempty subsequence with zero sum, then $f(S)\geq 2m-1$. This answers an open question formulated by Gao and Leader. They proved the same result with the restriction $(m,6)=1$.

On the Maximal Cross Number of Unique Factorization Zero-Sum Sequences over a Finite Abelian Group

Integers ◽  
2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Weidong Gao ◽  
Linlin Wang
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Unique Factorization ◽  
The Cross ◽  
Cross Number ◽  
Zero Sum

Abstract.Letdenote the cross number ofWe determine

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