Examining $H$-Closed Ducci Sequences on $\mathbb{Z}_m^n$

TL;DR

This work analyzes when Ducci sequences on $\mathbb{Z}_m^n$ are $H$-closed, i.e., invariant under rotations by $H^{\beta}$ for $-n<\beta<n$. It develops a framework using the Ducci map $D$, the rotation $H$ with $D=I+H$, the basic Ducci sequence, and coefficient arrays $a_{r,s}$ to derive exact $H$-closed results for $n=3$ in several modulus classes, and extends the investigation to even $n$ with prime moduli via a combination of algebraic and modular techniques. The paper also reports extensive computational exploration for broader $n,m$, identifying patterns and formulating conjectures (e.g., involving $\phi(m)$ and modulus factorization) that guide future proofs. These results contribute to understanding the cycle structure of Ducci sequences under symmetry and have potential implications for sequence dynamics on finite abelian groups.

Abstract

Let $D$ be an endomorphism on $\mathbb{Z}_m^n$ 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 the sequence $\{D^α(\mathbf{u})\}_{α=0}^{\infty}$ the Ducci sequence of $\mathbf{u} \in \mathbb{Z}_m^n$, which always enters a cycle. Now let $H$ be an endomorphism on $\mathbb{Z}_m^n$ such that \[H(x_1, x_2, ..., x_n)=(x_2, x_3, ..., x_n, x_1).\] In this paper, we will talk about a few cases when $\mathbf{u}$ and $H^β(\mathbf{u})$ have the same Ducci cycle for $β> 0$, as well as prove a few cases of $n,m$ where this is guaranteed for every $\mathbf{u} \in \mathbb{Z}_m^n$.

Figures (5)

• Figure 1: Transition Graph for $\mathbb{Z}_6^3$
• Figure 2: Transition Graph for $\mathbb{Z}_{10}^3$
• Figure 3: Cases of $n,m$ where $\mathbb{Z}_m^n$ is $H$-closed
• Figure 4: Cases of $n,m$ where $\mathbb{Z}_m^n$ is Weakly $H$-closed
• Figure 5: Cases of $n,m$ where $\mathbb{Z}_m^n$ is not Weakly $H$-closed

Theorems & Definitions (10)

• Definition 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Corollary 6
• proof : Proof of Theorem \ref{['HclosedTheorem']}:
• proof : Proof of Theorem \ref{['Hclosed_even']}