On the structure of $p$-zero-sum free sequences and its application to a variant of Erdos--Ginzburg--Ziv theorem

Abstract

Let $p$ be any odd prime number. Let $k$ be any positive integer such that $2\le k\le [\frac{p+1}{3}]+1$. Let $S=(a_1,a_2,...,a_{2p-k})$ be any sequence in ${\mathbb{Z}}_p$ such that there is no subsequence of length $p$ of $S$ whose sum is zero in $\zp$. Then we prove that we can arrange the sequence $S$ as follows: $S=(\underset{u\mathrm{times}}{\underbrace{a,a,...,a}},\underset{v\mathrm{times}}{\underbrace{b,b,>...,b}},a_1^{\prime },a_2^{\prime },>...,a_{2p-k-u-v}^{\prime })$ where $u\ge v$, $u+v\ge 2p-2k+2$ and $a-b$ generates $\zp$. This extends a result in \cite{gao10} to all primes $p$ and $k$ satisfying $(p+1)/4+3\le k\le (p+1)/3+1$. Also, we prove that if $g$ denotes the number of distinct residue classes modulo $p$ appearing in the sequence $S$ in $\zp$ of length $2p-k$ $(2\le k\le [(p+1)/4]+1)$, and $g\ge 2\sqrt{2}\sqrt{k-2}$, then there exists a subsequence of $S$ of length $p$ whose sum is zero in $\zp$.

11 pages