I try to understand Terence Tao's paper on the Collatz Conjecture [1909.03562], but got stuck on page 25.
We have $n$ copies of a geometric random variable of mean $2$, denoted by $a_i$ and $a_{[i,j]}$ is defined to be the sum over them from $a_i$ to $a_j$. It is then claimed, that if
$$|a_{[i,j]}-2(j-i)| \leq C_A(\sqrt{(j-i)(\log(n))}+\log(n))$$
holds for all $i,j$, that then we have
$$a_{[1,n]} \geq 2n-C_A(\sqrt{n(\log(n))}+\log(n)) > n \frac{\log 3}{\log 2}$$
with large $n$.
I see that I get at least
$$a_{[1,n]} \geq 2(n-1)-C_A(\sqrt{n(\log(n))}+\log(n))$$
which had the same consequence, but is this a typo or can I get even the stronger statement?
But the more important question is the following. He introduces a stopping time $k_{\text{stop}}$ with the property
$$a_{[1,k_{\text{stop}}]} \leq n \frac{\log 3}{\log 2} - C_A^2 \log(n)<a_{[1, k_{\text{stop}}+1]}$$
It is then claimed, that
$$k_{\text{stop}}= n \frac{\log(3)}{2 \log(2)}+O(C_A^2 \log(n))$$
I do not understand the last statement. In the "worst" case, all the $a_i$ are 1 and then this would not hold. Clearly, this example would violate the inequality in the beginning, but why is this the case in general?
Furthermore, he claims, that the stopping time $l$ iff
$$a_{[1,l]} \leq n \frac{\log 3}{\log 2}-C_A^3 \log n < a_{[1,l+1]}$$
Where does the $C_A^3$ instead of $C_A^2$ come from?
Thank you for your suggestion writing Terence Tao, he answered my question on his blog.
Just for the case, someone else faces the same questions in future: https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/#comment-570769