On the cluster structures in Collatz level sets

TL;DR

This paper investigates persistent cluster structures within the Collatz tree's level sets, linking their stability to a specific orbit steadiness condition that remains consistent across all levels.

Contribution

It introduces a new perspective on the Collatz tree by connecting cluster structures to orbit steadiness, providing conditions for their persistence across levels.

Findings

Cluster structures are maintained across all levels under certain conditions.

Orbit steadiness is key to the persistence of these structures.

Provides a mathematical criterion for cluster stability in the Collatz tree.

Abstract

The cluster structures that can be observed in the first few level sets of the Collatz tree are maintained through all its levels, provided that the orbit steadiness \[ \prod_{\substack{k \in R(n)\\ k \equiv 4\ (\mathrm{mod}\ 6)}} \frac{k-1}k \] of the elements $n$ of the Collatz tree is suitably bounded from below, where $R(n)$ denotes the Collatz orbit of $n$.