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

Czasopismo Techniczne ◽

10.37705/techtrans/e2020027 ◽

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.

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On zero-sum subsequences of prescribed length

International Journal of Number Theory ◽

10.1142/s1793042118500112 ◽

2017 ◽

Vol 14 (01) ◽

pp. 167-191 ◽

Cited By ~ 3

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.

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On the generalized Davenport constant and the Noether number

Open Mathematics ◽

10.2478/s11533-013-0259-z ◽

2013 ◽

Vol 11 (9) ◽

Cited By ~ 1

Author(s):

Kálmán Cziszter ◽

Mátyás Domokos

Keyword(s):

Finite Group ◽

Abelian Group ◽

Finite Abelian Group ◽

Davenport Constant ◽

Polynomial Invariants ◽

Arbitrary Finite Group ◽

Zero Sum ◽

Zero Vector ◽

Degree Bound

AbstractKnown results on the generalized Davenport constant relating zero-sum sequences over a finite abelian group are extended for the generalized Noether number relating rings of polynomial invariants of an arbitrary finite group. An improved general upper degree bound for polynomial invariants of a non-cyclic finite group that cut out the zero vector is given.

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On some weighted zero-sum constants

International Journal of Number Theory ◽

10.1142/s1793042117500191 ◽

2017 ◽

Vol 13 (02) ◽

pp. 301-308 ◽

Cited By ~ 2

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.

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

The Electronic Journal of Combinatorics ◽

10.37236/620 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 3

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.

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

International Journal of Number Theory ◽

10.1142/s1793042114500481 ◽

2014 ◽

Vol 10 (07) ◽

pp. 1637-1647 ◽

Cited By ~ 1

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].

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On some weighted zero-sum constants II

International Journal of Number Theory ◽

10.1142/s1793042118500264 ◽

2018 ◽

Vol 14 (02) ◽

pp. 383-397 ◽

Cited By ~ 1

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].

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Zero-sum subsequences of distinct lengths

International Journal of Number Theory ◽

10.1142/s1793042115500931 ◽

2015 ◽

Vol 11 (07) ◽

pp. 2141-2150 ◽

Cited By ~ 2

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).

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

Periodica Mathematica Hungarica ◽

10.1007/s10998-021-00418-6 ◽

2021 ◽

Author(s):

Weidong Gao ◽

Siao Hong ◽

Wanzhen Hui ◽

Xue Li ◽

Qiuyu Yin ◽

...

Keyword(s):

Abelian Group ◽

Finite Abelian Group ◽

Zero Sum

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On Subsequence Sums of a Zero-sum Free Sequence

The Electronic Journal of Combinatorics ◽

10.37236/970 ◽

2007 ◽

Vol 14 (1) ◽

Cited By ~ 5

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 $(m,6)=1$.

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