scholarly journals The Weighted Davenport Constant of a Group and a Related Extremal Problem

10.37236/7996 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Niranjan Balachandran ◽  
Eshita Mazumdar
Keyword(s):  
Abelian Group ◽  
Extremal Problem ◽  
Prime Order ◽  
Finite Abelian Group ◽  
Arbitrary Sequence ◽  
Davenport Constant ◽  
Optimal Bounds ◽  
Empty Subset ◽  
Related Extremal Problem ◽  
Well Posed

For a finite abelian group $G$ written additively, and a non-empty subset $A\subset [1,\exp(G)-1]$ the weighted Davenport Constant of $G$  with respect to the set $A$, denoted $D_A(G)$, is the least positive integer $k$ for which the following holds: Given an arbitrary sequence $(x_1,\ldots,x_k)$ with $x_i\in G$, there exists a non-empty subsequence  $(x_{i_1},\ldots,x_{i_t})$ along with $a_{j}\in A$ such that $\sum_{j=1}^t a_jx_{i_j}=0$. In this paper, we pose and study a natural new extremal problem that arises from the study of $D_A(G)$:  For an integer $k\ge 2$, determine $f^{(D)}_G(k):=\min\{|A|: D_A(G)\le k\}$ (if the problem posed makes sense). It turns out that for $k$ 'not-too-small', this is a well-posed problem and one of the most interesting cases occurs for $G=\mathbb{Z}_p$, the cyclic group of prime order, for which we obtain near optimal bounds for all $k$ (for sufficiently large primes $p$), and asymptotically tight (up to constants) bounds for $k=2,4$.

scholarly journals Remarks on the plus–minus weighted Davenport constant

2014 ◽  
Vol 10 (05) ◽  
pp. 1219-1239 ◽  
Author(s):  
Luz E. Marchan ◽  
Oscar Ordaz ◽  
Wolfgang A. Schmid
Keyword(s):  
Abelian Group ◽  
Lower Bounds ◽  
Finite Abelian Group ◽  
Davenport Constant ◽  
Empty Subset

For (G, +) a finite abelian group the plus–minus weighted Davenport constant, denoted D±(G), is the smallest ℓ such that each sequence g1…gℓover G has a weighted zero-subsum with weights +1 and -1, i.e. there is a non-empty subset I ⊂ {1,…,ℓ} such that ∑i∈Iaigi= 0 for ai∈ {+1, -1}. We present new bounds for this constant, mainly lower bounds, and also obtain the exact value of this constant for various additional types of 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 (k, l)-sets in cyclic groups of odd prime order

2001 ◽  
Vol 63 (1) ◽  
pp. 115-121 ◽  
Author(s):  
T. Bier ◽  
A. Y. M. Chin
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Arithmetic Progression ◽  
Prime Order ◽  
Finite Abelian Group ◽  
Maximum Cardinality ◽  
Cyclic Groups ◽  
Positive Integers

Let A be a finite Abelian group written additively. For two positive integers k, l with k ≠ l, we say that a subset S ⊂ A is of type (k, l) or is a (k, l) -set if the equation x1 + x2 + … + xk − xk+1−… − xk+1 = 0 has no solution in the set S. In this paper we determine the largest possible cardinality of a (k, l)-set of the cyclic group ℤP where p is an odd prime. We also determine the number of (k, l)-sets of ℤp which are in arithmetic progression and have maximum cardinality.

scholarly journals On the generalized Davenport constant and the Noether number

2013 ◽  
Vol 11 (9) ◽  
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.

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.

scholarly journals A Weighted Generalization of Gao's n + D − 1 Theorem

2008 ◽  
Vol 17 (6) ◽  
pp. 793-798 ◽  
Author(s):  
YAHYA O. HAMIDOUNE
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Davenport Constant ◽  
Image Position

Let G denote a finite abelian group of order n and Davenport constant D, and put m = n + D − 1. Let x = (x1,. . .,xm) ∈ Gm. Gao's theorem states that there is a reordering (xj1, . . ., xjm) of x such that Let w = (x1, . . ., wm) ∈ ℤm. As a corollary of the main result, we show that there are reorderings (xj1, . . ., xjm) of x and (wk1, . . ., wkm) of w, such that where xj1 is the most repeated value in x. For w = (1, . . ., 1), this result reduces to Gao's theorem.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

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
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