Zero sums in restricted sequences

Abstract

A sequence $\bfx=(x_1,\ldots,x_m)$ of elements of ${\mathbb{Z}}_n$ 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 ${\sum }_i{a}_i{x}_i=0$. Here $\bfzero_m=(0,\ldots,0)\in\Z_n^m$. Similarly, the sequence $\bfx$ is called an \textit{$A$-weighted Erdős Z-sequence} if there exists $\bfa:=(a_1,\ldots,a_m)\in (A\cup\{0\})^m\setminus\{\bfzero_m\}$ with $|Supp(\bfa)|=n$, such that ${\sum }_i{a}_i{x}_i=0$, where $Supp(\bfa):=\{i: a_i\ne 0\}$. A ${\mathbb{Z}}_n$-sequence $\bfx$ is called $k$-restricted if no element of ${\mathbb{Z}}_n$ appears more than $k$ times in $\bfx$. In this paper, we study the problem of determining the least value of $m$ for which a $k$-restricted ${\mathbb{Z}}_n$-sequence of length $m$ is an $A$-weighted Davenport Z-sequence (resp. an$A$-weighted Erdős Z-sequence). We also consider the same problem for random ${\mathbb{Z}}_n$ sequences, for certain very natural choices for the set $A$.