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

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 Colla