An approximation of the Collatz map and a lower bound for the average total stopping time

Define the map $\mathsf{T}$ on the positive integers by $\mathsf{T}(m)=\frac{m}{2}$ if $m$ is even and by $\mathsf{T}(m)=\frac{3m+1}{2}$ if $m$ is odd. Results of Terras and Everett imply that, given any $\epsilon>0$, almost all $m\in\mathbb{Z}^+$ (in the sense of natural density) fulfill $(\frac{\sqrt{3}}{2})^km^{1-\epsilon}\leq \mathsf{T}^k(m)\leq (\frac{\sqrt{3}}{2})^km^{1+\epsilon}$ simultaneously for all $0\leq k\leq \alpha\log m$ with $\alpha=(\log 2)^{-1}\approx 1.443$. We extend this result to $\alpha=2(\log\frac{4}{3})^{-1}\approx 6.952$, which is the maximally possible value. Set $\mathsf{T}_{\min}(m):=\min_{n\in\mathbb{N}}\mathsf{T}^n(m)$. As an immediate consequence, one has $\mathsf{T}_{\min}(m)\leq\mathsf{T}^{\left\lfloor2(\log\frac{4}{3})^{-1}\log m\right\rfloor}(m)\leq m^{\epsilon}$ for almost all $m\in\mathbb{Z}^+$ for any given $\epsilon>0$. Previously, Korec has shown that $\mathsf{T}_{\min}(m)\leq m^\epsilon$ for almost all $m\in\mathbb{Z}^+$ if $\epsilon>\frac{\log3}{\log4}$, and recently Tao proved that $\mathsf{T}_{\min}(m)\leq f(m)$ for almost all $m\in\mathbb{Z}^+$ (in the sense of logarithmic density) for all functions $f$ diverging to $\infty$. Denote by $\tau(m)$ the minimal $n\in\mathbb{N}$ for which $\mathsf{T}^n(m)=1$ if there exists such an $n$ and set $\tau(m)=\infty$ otherwise. As another application, we show that $\liminf_{x\rightarrow\infty}\frac{1}{x\log x}\sum_{m=1}^{\lfloor x\rfloor}\tau(m)\geq 2(\log\frac{4}{3})^{-1}$, partially answering a question of Crandall and Shanks. Under the assumption that the Collatz Conjecture is true in the strong sense that $\tau(m)$ is in $O(\log m)$, we show that $\lim_{x\rightarrow\infty}\frac{1}{x\log x}\sum_{m=1}^{\lfloor x\rfloor}\tau(m)= 2(\log\frac{4}{3})^{-1}$.