Weighted Sequences in Finite Cyclic Groups

TL;DR

This paper investigates weighted subsequence sums in finite cyclic groups, proving a permutation existence under certain multiplicity constraints, and relates to conjectures in additive combinatorics like Erdős-Ginzburg-Ziv.

Contribution

It establishes a new permutation result for weighted sequences in cyclic groups with bounded multiplicities, advancing understanding of weighted sum problems.

Findings

Proves permutation existence for sequences with bounded multiplicity

Connects results to Bialostocki's conjecture and Erdős-Ginzburg-Ziv theorem

Provides conditions under which weighted sums cover the entire group

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.