Related papers: $Q_p$-weighted zero-sum constants

Let $A,B\subseteq \mathbb Z_n\setminus\{0\}$. A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_n$ is called an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that…

Let $A\subseteq \mathbb Z_n$ be a subset. A sequence $S=(x_1,\ldots,x_k)$ in $\mathbb Z_n$ is said to be an $A$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ such that $a_1x_1+\cdots+a_kx_k=0$. By a square, we shall mean a…

Let $A,B\subseteq\mathbb Z_n$ be given and $S=(x_1,\ldots, x_k)$ be a sequence in $\mathbb Z_n$. We say that $S$ is an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that…

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.…

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…

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 a subsequence of consecutive terms whose $A$-weighted sum is zero, where the weight set $A\subseteq \mathbb…

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…

A zero-sum sequence of integers is a sequence of nonzero terms that sum to 0. Let $k>0$ be an integer and let $[-k,k]$ denote the set of all nonzero integers between $-k$ and $k$. Let $\ell(k)$ be the smallest integer $\ell$ such that any…

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…

It is shown that there is a constant A and a density one subset S of the positive integers, such that for all q in S there is some 1<=p<q, (p, q)=1, so that p/q has all its partial quotients bounded by A.

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…

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…

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.…

A zero-sum sequence over ${\mathbb Z}$ is a sequence with terms in ${\mathbb Z}$ that sum to $0$. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ${\mathbb Z}$ with…

Let $G$ be a finite additive abelian group. For given $k$ a positive integer, the $k$-Harborth constant $g^k(G)$ is defined to be the smallest positive integer $t$ such that given a set $S$ of elements of $G$ with size $t$ there exists a…

Let $q$ be a power of a prime $p$, and let $V$ be an $n$-dimensional space over the field GF$(q)$. A $Z_p$-valued function $C$ on the set of $k$-dimensional subspaces of $V$ is called a $k$-uniform $Z_p$-null design of strength $t$ if for…

We consider two families of weighted zero-sum constants for finite abelian groups. For a finite abelian group $( G , + )$, a set of weights $W \subset \mathbb{Z}$, and an integral parameter $m$, the $m$-wise Davenport constant with weights…

For a finite abelian group $(G,+)$ the Harborth constant is defined as the smallest integer $\ell$ such that each squarefree sequence over $G$ of length $\ell$ has a subsequence of length equal to the exponent of $G$ whose terms sum to $0$.…

Let $\{Z_t, t\geq 0\}$ be a strictly stable process on $\R$ with index $\alpha\in (0,2]$. We prove that for every $p > \alpha$, there exists $\gamma = \gamma (\alpha, p)$ and $\k = \k (\alpha, p)\in (0, +\infty)$ such that…

Let $G$ be a finite additive abelian group with exponent $d^kn, d,n>1,$ and $k$ a positive integer. For $S$ a sequence over $G$ and $A=\{1,2,\ldots,d^kn-1\}\setminus\{d^kn/d^i:i\in[1,k]\}, $ we investigate the lower bound of the number…