Zero sums in restricted sequences

TL;DR

This paper investigates the minimal length of restricted sequences over cyclic groups that guarantee the existence of certain weighted zero-sum subsequences, extending classical zero-sum problems with restrictions and randomness.

Contribution

It introduces new bounds for the minimal sequence length ensuring weighted zero-sum properties in restricted and random sequences over cyclic groups.

Findings

Established bounds for the minimal length of $k$-restricted sequences as weighted zero-sum sequences.

Analyzed the problem for specific sets $A$ and random sequences.

Extended classical zero-sum results to restricted and probabilistic contexts.

Abstract

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 $\bfzero_m=(0,\ldots,0)\in\Z_n^m$. Similarly, the sequence $\bfx$ is called an \textit{$A$-weighted Erd\H{o}s 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 $\sum_i a_ix_i=0$, where $Supp(\bfa):=\{i: a_i\ne 0\}$. A $\Z_n$-sequence $\bfx$ is called $k$-restricted if no element of $\Z_n$ 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 $\Z_n$-sequence of length $m$ is an $A$-weighted Davenport Z-sequence (resp. an$A$-weighted Erd\H{o}s Z-sequence). We also consider the same problem for random $\Z_n$ sequences, for certain very natural choices for the set $A$.