Almost-Boundedness of Collatz-Type Maps on Number Rings via Archimedean Methods

Abstract

Let $p$ be an integer $\geq2$, not necessarily prime, and let $K$ be a number field. In 2019, Tao showed that almost every positive integer (in the sense of logarithmic density) had an almost bounded trajectory under the Collatz map. In this paper, we generalize this result to a large class of Collatz-type maps ($p$-Hydras) acting on the ring of integers $\mathcal{O}_{K}$ of a given number field $K$ of dimension $d$ over $\mathbb{Q}$. Given an ideal $Λ\subseteq\mathcal{O}_{K}$ of index $p$, a $p$-Hydra map $H:\mathcal{O}_{K}\rightarrow\mathcal{O}_{K}$ is a collection of affine-linear maps $\left\{ H_{j}:K\rightarrow K\right\} _{j\in\mathcal{O}_{K}/Λ}$ of the form $z\mapsto r_{j}z+c_{j}$ for constants $c_{j}\in K$ and invertible $\mathbb{Q}$-linear maps $r_{j}:K\rightarrow K$ so that the map $1-r_{0}$ is invertible and so that $H\left(z\right)$ is defined to be $H_{\left[z\right]_Λ}\left(z\right)$, where $\left[z\right]_Λ$ is the projection of $z$ mod $Λ$. We show that the direct analogue of Tao's almost-boundedness result holds provided, in addition to several minor conditions, that the Dimension Constraint $\left(p-1\right)^{2}ρ_{H}^{d}<1$ is satisfied, where $ρ_{H}=\prod_{j=0}^{p-1}\left\Vert r_{j}\right\Vert _{\textrm{Mink}}$ is the product of the operator norms of the $r_{j}$s with respect to the Minkowski embedding of $K$. Unlike Tao's original argument, which uses a $3$-adic Fourier decay result for the Syracuse Random Variables, our approach employs analysis of an archimedean flavor, and does not require any Fourier decay estimates for our generalized Syracuse Random Variables. Our result includes Tao's original result as a special case.