关于 $\mathbb{F}_2[x]$ 中 Collatz 映射的停时

A General Class of Collatz Sequence and Ruin Problem

In this paper we show the probabilistic convergence of the original Collatz (3n + 1) (or Hotpo) sequence to unity. A generalized form of the Collatz sequence (GCS) is proposed subsequently. Unlike Hotpo, an instance of a GCS can converge to integers other than unity. A GCS can be generated using the concept of an abstract machine performing arithmetic operations on different numerical bases. Original Collatz sequence is then proved to be a special case of GCS on base 2. The stopping time of GCS sequences is shown to possess remarkable statistical behavior. We conjecture that the Collatz conver…

Density of Collatz Trajectories

In this paper I present a new method of studying the densities of the Collatz trajectories generated by a set $S \subset \mathbb{N}$. This method is used to furnish an alternative proof that $d(\{y \in \mathbb{N} : \exists k \text{ where } T^k(y) < cy\}) = 1$ for all $c > 0$. Finally, I briefly discuss how the ideas presented in this paper could be used to improve the result that $d(\{y \in \mathbb{N} : \exists k \text{ where } T^k(y) < y^c\}) = 1$ for $c > \log_4 3$.

关于 $\mathbb{F}_2[x]$ 上 Collatz 映射的平均停时

On the average stopping time of the Collatz map in $\mathbb{F}_2[x]$

本文研究了 $\mathbb{F}_2[x]$ 上 Collatz 映射的平均停时，证实了 $\lim_{n\to\infty}\rho_1(n)/n=2$ 的猜想，并给出了停时上界 $O(\text{deg}(f)^{1.5})$ 的新证明。

多项式环与完备化中的 Collatz 映射类似物

The Collatz map analogue in polynomial rings and in completions

本文在任意交换环的多项式环及形式幂级数环上研究 Collatz 映射类似物，证明正特征情形下所有多项式最终周期，并给出有限环形式幂级数及 2-进整数上环圈数的计数公式。

实数域上考拉兹映射类似物的几乎所有轨道都能达到有界值

Almost all orbits of an analogue of the Collatz map on the reals attain bounded values

该研究定义了实数域上的考拉兹类似映射$C_\mathbb{R}$，证明其几乎所有正实数轨道的下极限有界，还刻画了迭代增长速度符合特定幂律范围的大集合性质。

Integer Representations and Trajectories of the 3x+1 Problem

This paper studies certain trajectories of the Collatz function. I show that if for each odd number $n$, $n\sim 3n+2$ then every positive integer $n \in \mathbb{N}\setminus 2^{\mathbb{N}}$ has the representation $$n=\left(2^{a_{k+1}}-\sum_{i=0}^{k}{2^{a_i}3^{k-i}}\right)/ 3^{k+1}$$ where $0\le a_0 \le a_1 \le \cdot \cdot \cdot \le a_{k+1}$. As a consequence, in order to prove Collatz Conjecture I illustrate that it is sufficient to prove $n\sim 3n+2$ for any odd $n\in \mathbb{N}\setminus 2^{\mathbb{N}} $. This is the main result of the paper.