On the cluster structures in Collatz level sets

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 $n$.

In v1 the terms "preimage" and "level set" were mixed up and used inconsistently. v2 corrects the definition and talks only about level sets