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 $2\leq k\leq [\frac{p+1}3]+1$. Let $S = (a_1,a_2,...,a_{2p-k})$ be any sequence in ${\Bbb Z}_p$ 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 = (\underbrace{a, a, ..., a}_{u {\rm times}}, \underbrace{b, b, >..., b}_{v {\rm times}}, a_1', a_2', >..., a_{2p-k-u-v}') $ where $u\geq v$, $u+v\geq 2p-2k+2$ and $a-b$ generates $\zp$. This extends a result in \cite{gao10} to all primes $p$ and $k$ satisfying $(p+1)/4+3\leq k\leq (p+1)/3+1$. Also, we prove that if $g$ denotes the number of distinct residue classes modulo $p$ appearing in the sequence $S$ in $\zp$ of length $2p-k$ $(2\leq k\leq [(p+1)/4]+1)$, and $g\geq 2\sqrt{2}\sqrt{k-2}$, then there exists a subsequence of $S$ of length $p$ whose sum is zero in $\zp$.