Related papers: Inverse Zero-Sum Problems III

Let $n$ be a positive integer and let $S$ be a sequence of $n$ integers in the interval $[0,n-1]$. If there is an $r$ such that any nonempty subsequence with sum $\equiv 0$ $\pmod n$ has length $=r,$ then $S$ has at most two distinct…

Let $\mathbb Z_n$ be the cyclic group of order $n \ge 3$ additively written. S. Savchev \& F. Chen (2007) proved that for each zero-sum free sequence $S = a_1 \bullet \dots \bullet a_t$ over $\mathbb Z_n$ of length $t > n/2$, there is an…

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of…

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…

Let $G$ be a finite abelian group. For any positive integers $d$ and $m$, let $\varphi_G(d)$ be the number of elements in $G$ of order $d$ and $\mathsf M(G,m)$ be the set of all zero-sum sequences of length $m$. In this paper, for any…

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be…

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(x_1g)\cdot\ldots\cdot(x_lg)$ where $g\in G$ and $x_1, \ldots, x_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be…

Let $G$ be a finite group. A sequence over $G$ is a finite multiset of elements of $G$, and it is called product-one if its terms can be ordered so that their product is the identity of $G$. The large Davenport constant $\D(G)$ is the…

Let $G$ be a group and $A\subseteq [1,\exp(G)-1]$. We define the constant ${\sf C}_A(G),$ which is the least positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has an $A$-weighted consecutive product-one…

Let $G$ be a finite abelian group, and let $S$ be a sequence over $G$. 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 determine all the sequences $S$…

The small Davenport constant ${\mathsf{d}}(G)$ of a finite group $G$ is defined to be the maximal length of a sequence over $G$ which has no non-trivial product-one subsequence. In this paper, we prove that ${\mathsf{d}}(G) = 6$ for the…

Let $G$ be an additive finite abelian group with exponent $\exp(G)$. For $L\subseteq \mathbb N$, let $\mathsf{s}_{L}(G)$ be the smallest integer $\ell$ such that every sequence $S$ over $G$ of length $\ell$ has a zero-sum subsequence $T$ of…

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…

In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds…

For a finite abelian group $G$ and a splitting field $K$ of $G$, let $d(G, K)$ denote the largest integer $l \in \N$ for which there is a sequence $S = g_1 \cdot ... \cdot g_l$ over $G$ such that $(X^{g_1} - a_1) \cdot ... \cdot (X^{g_l} -…

We show that a finite zero-sum-free sequence $\alpha$ over an abelian group has at least $c|\alpha|^{4/3}$ distinct subsequence sums, unless $\alpha$ is "controlled" by a small number of its terms; here $|\alpha|$ denotes the number of…

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 an additive abelian group. A sequence $S=g_1\cdot\ldots\cdot g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are $\varepsilon_1,\ldots,\varepsilon_{\ell}\in\{-1,1\}$ such that $\varepsilon_1…

Let $G$ be a finite abelian group and let $A\subseteq \mathbb{Z}$ be nonempty. Let $D_A(G)$ denote the minimal integer such that any sequence over $G$ of length $D_A(G)$ must contain a nontrivial subsequence $s_1... s_r$ such that…

We determine the exact value of the $\eta$-constant and the multiwise Davenport constants for finite abelian groups of rank three having the form $G \simeq C_2 \oplus C_{n_2} \oplus C_{n_3}$ with $2 \mid n_2 \mid n_3$. Moreover, we…