Related papers: Davenport constant with weights

For two positive integers $m$ and $M$, we study the Davenport constant of the interval of integers $[\![ -m,M ]\!]$, that is the maximal length of a minimal zero-sum sequence composed of elements from $[\![ -m,M ]\!]$. We prove the…

For a finite abelian group $(G,+)$, the constant $C(G)$ is defined to be the smallest natural number $k$ such that any sequence in $G$ having length $k$ will have a subsequence of consecutive terms whose sum is zero. For a subset…

Let $\mathcal{S}$ be a finite commutative semigroup written additively, and let $\exp(\mathcal{S})$ be its exponent which is defined as the least common multiple of all periods of the elements in $\mathcal{S}$. For every sequence $T$ of…

For a finite abelian group $G$ and a positive integer $d$, let $\mathsf s_{d \mathbb N} (G)$ denote the smallest integer $\ell \in \mathbb N_0$ such that every sequence $S$ over $G$ of length $|S| \ge \ell$ has a nonempty zero-sum…

Let $C_2$ be the cyclic group of order $2$ and $D_{2n}$ be the dihedral group of order $2n$, where $n$ is even. In this paper, we provide the exact values of some zero-sum constants over $D_{2n} \times C_2$, namely small Davenport constant,…

For a finite abelian group $G$ with $\exp(G)=n$ and an integer $k\ge 2$, Balachandran and Mazumdar \cite{BM} introduced the extremal function $\fD_G(k)$ which is defined to be $\min\{|A|: \emptyset \neq A\subseteq[1,n-1]\textrm{\ with\…

The Erd\H{o}s-Burgess constant of a semigroup $S$ is the smallest positive integer $k$ such that any sequence over $S$ of length $k$ contains a nonempty subsequence whose elements multiply to an idempotent element of $S$. In the case where…

The Harborth constant of a finite group $G$ is the smallest integer $k\geq \exp(G)$ such that any subset of $G$ of size $k$ contains $\exp(G)$ distinct elements whose product is $1$. Generalizing previous work on the Harborth constants of…

Let $G$ be a finite abelian group of exponent $n$, written additively, and let $A$ be a subset of $\mathbb{Z}$. The constant $s_A(G)$ is defined as the smallest integer $\ell$ such that any sequence over $G$ of length at least $\ell$ has an…

Let $G$ be a multiplicatively written finite group of order $n$. The Erd\H{o}s-Ginzburg-Ziv Theorem constant of the group $G$, denoted $\mathsf E(G)$, is defined as the smallest positive integer $\ell$ with the following property: for any…

We generalize the Davenport transform and use it to prove that, for a (possibly non-commutative) cancellative semigroup $\mathbb A = (A, +)$ and non-empty subsets $X,Y$ of $A$ such that the subsemigroup generated by $Y$ is commutative, we…

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. For $k\in \mathbb N$, let $\mathcal U_k(H)$ denote the set of all $m\in \mathbb N$ with the following property: There exist atoms $u_1,…

A sequence $\bfx=(x_1,\ldots,x_m)$ of elements of $\Z_n$ is called an \textit{$A$-weighted Davenport Z-sequence} if there exists $\bfa:=(a_1,\ldots,a_m)\in (A\cup\{0\})^m\setminus\bfzero_m$ such that $\sum_i a_ix_i=0$. Here…

Let $G$ be a multiplicatively written finite group. We denote by $\mathsf E(G)$ the smallest integer $t$ such that every sequence of $t$ elements in $G$ contains a product-one subsequence of length $|G|$. In 1961, Erd\H{o}s, Ginzburg and…

Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at…

For $A\subseteq \mathbb Z_n$, the $A$-weighted Gao constant $E_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of length $n$, whose $A$-weighted sum is zero.…

Let $G$ be a multiplicative finite group and $S=a_1\cdot\ldots\cdot a_k$ a sequence over $G$. We call $S$ a product-one sequence if $1=\prod_{i=1}^ka_{\tau(i)}$ holds for some permutation $\tau$ of $\{1,\ldots,k\}$. The small Davenport…

This article focuses on the study of zero-sum invariants of finite non-abelian groups. We address two main problems: the first centers on the ordered Davenport constant and the second on Gao's constant. We establish a connection between the…

Let $\F_q[x]$ be the ring of polynomials over the finite field $\F_q$, and let $f$ be a polynomial of $\F_q[x]$. Let $R=\frac{\F_q[x]}{(f)}$ be a quotient ring of $\F_q[x]$ with $0\neq R\neq \F_q[x]$. Let $\mathcal{S}_R$ be the…

The Harborth constant of a finite group $G$, denoted $\gs(G)$, is the smallest integer $k$ such that the following holds: For $A\subseteq G$ with $|A|=k$, there exists $B\subseteq A$ with $|B|=\exp(G)$ such that the elements of $B$ can be…