Related papers: On the Number of Weighted Zero-sum Subsequences

Let $G=C_n\oplus C_n$ and let $k\in [0,n-1]$. We study the structure of sequences of terms from $G$ with maximal length $|S|=2n-2+k$ that fail to contain a nontrivial zero-sum subsequence of length at most $2n-1-k$. For $k\leq 1$, this is…

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

Let {\cal G}=(G,w) be a positive-weighted simple finite graph, that is, let G be a simple finite graph endowed with a function w from the set of the edges of G to the set of the positive real numbers. For any subgraph G' of G, we define…

In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which…

Let $G$ be a finite abelian group and $D(G)$ denote the Davenport constant of $G$. We derive new upper bound for the Davenport constant for all groups of rank three. Our main result is that: $$D(C_{n_1}\oplus C_{n_2}\oplus C_{n_3})\le…

Let $G$ be a finite group. The small Davenport constant $\mathsf d(G)$ of $G$ is the maximal integer $\ell$ such that there is a sequence of length $\ell$ over $G$ which has no nonempty product-one subsequence. In 2007, Bass conjectured…

Let $G$ be an additive finite abelian group and $\Gamma \subset \operatorname{End} (G)$ be a subset of the endomorphism group of $G$. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ over $G$ is a ($\Gamma$-)weighted zero-sum sequence if…

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 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. Then, $\so(G)$ and $\eta(G)$ denote the smallest integer $\ell$ such that each sequence over $G$ of length at least $\ell$ has a subsequence whose terms sum to $0$ and whose length is equal to and at…

Given $A\subseteq\mathbb Z_n$, the constant $C_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum subsequence having consecutive terms. The value of…

For a sequence $S$ over a finite abelian group, let $MZ(S)$ denote the length of the shortest nonempty zero-sum subsequence of $S$. We prove that if $G$ is finite abelian of order $n$ and $S$ has length $n$, then $MZ(S)\le n-|\supp(S)|+1$.…

Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$. We prove that $d(G)\leq \frac{|G|}{p}+9p^2-10p$, where $d(G)$ denotes the small Davenport constant of $G$ which is defined as…

Let $G$ be a finite group written multiplicatively. By a sequence over $G$, we mean a finite sequence of terms from $G$ which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be…

For a finite set $A\subset\mathbb{N}$ and $k\in \mathbb{N}$, let $\omega_k(A) = \sum_{i\in A, i\neq k}1$. For each $n\in \mathbb{N}$, define $$a_{k, n}\ =\ |\{E\subset \mathbb{N}\,:\, E = \emptyset\mbox{ or } \omega_k(E) < \min E\leqslant…

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is…

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $d$ such that every sequence of $G$ with $d$ elements has a non-empty subsequence with product $1$. Let $C_n \simeq \mathbb…

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 $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\…

For every even integer N, denote by D_{1,2}(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D_{1,2}(N).