On the maximal multiplicity of long zero-sum free sequences over $C_p\oplus C_p$

TL;DR

This paper improves understanding of the structure of long zero-sum free sequences over the group C_p ⊕ C_p, showing that such sequences must contain an element with high multiplicity when their length exceeds a certain threshold.

Contribution

It introduces a modified method to establish a lower bound on the maximum multiplicity of elements in long zero-sum free sequences over C_p ⊕ C_p for large primes.

Findings

Sequences longer than 2p - c√p contain an element repeated at least p^{1/4 - ε} times.

The method adapts previous techniques to handle larger sequence lengths.

Results hold for sufficiently large primes depending on ε and c.

Abstract

In this paper, we point out that the method used in [Acta Arith. 128(2007) 245-279] can be modified slightly to obtain the following result. Let $\varepsilon \in (0,\frac 14)$ and $c>0$, and let $p$ be a sufficiently large prime depending on $\varepsilon$ and $c$. Then every zero-sumfree sequence $S$ over $C_p\oplus C_p$ of length $|S|\geq 2p-c\sqrt{p}$ contains some element at least $\lfloor p^{\frac14-\varepsilon}\rfloor$ times.