Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$

TL;DR

This paper investigates the possible cycle lengths of sequences generated by a specific Ducci function on finite cyclic groups, identifying conditions that lead to shorter cycles than the maximum possible.

Contribution

It characterizes the range of Ducci cycle lengths on  groups and provides conditions for shorter cycles, extending understanding of these sequences' periodic behavior.

Findings

Identifies all possible cycle lengths of Ducci sequences on  groups.

Provides criteria for when a sequence's period is less than the maximum.

Analyzes the structure of Ducci cycles in modular settings.

Abstract

Let $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call $D$ the Ducci function and the sequence $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ the Ducci sequence of $\mathbf{u}$ for $\mathbf{u} \in \mathbb{Z}_m^n$. Every Ducci sequence enters a cycle, so we can let $\text{Per}(\mathbf{u})$ be the number of tuples in the Ducci cycle of $\mathbf{u}$, or the period of $\mathbf{u}$. In this paper, we will look at what different possible values of $\text{Per}(\mathbf{u})$ we can have and some conditions that if $\mathbf{u}$ meets at least one of them, $\mathbf{u}$ will generate a period smaller than the maximum period.