Weighted Sequences in Finite Cyclic Groups

Abstract

Let $p>7$ be a prime, let $G=\Z/p\Z$, and let $S_1=\prod_{i=1}^p g_i$ and $S_2=\prod_{i=1}^p h_i$ be two sequences with terms from $G$. Suppose that the maximum multiplicity of a term from either $S_1$ or $S_2$ is at most $\frac{2p+1}{5}$. Then we show that, for each $g\in G$, there exists a permutation $\sigma$ of $1,2,..., p$ such that $g=\sum_{i=1}^{p}(g_i\cdot h_{\sigma(i)})$. The question is related to a conjecture of A. Bialostocki concerning weighted subsequence sums and the Erd\H{o}s-Ginzburg-Ziv Theorem.