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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?

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    $\begingroup$ Please link to the abstract, rather than the PDF file. Some people may want to try the appetiser before ordering the main course. $\endgroup$ Commented Jul 1, 2020 at 12:42
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    $\begingroup$ why not ask the author on his website (under the post discussing this result) as he is very good at answering legitimate questions about his posts terrytao.wordpress.com/2019/09/10/… $\endgroup$
    – Conrad
    Commented Jul 1, 2020 at 12:54
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    $\begingroup$ @Conrad Tao's time is extremely valuable. Why disturb the gods before giving mere mortals a chance to answer the question? $\endgroup$ Commented Jul 1, 2020 at 14:03
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    $\begingroup$ @Rodrigo - this is a preprint so I think that pertinent questions and comments are always useful and appreciated and as noted T Tao answers pretty much all pertinent questions on his blog - imho this is not a waste of time both from the pedagogical point of view (as a math professor one always wants to get feedback, as things that seem quite obvious to one, well...) and the research one (T Tao's papers, books, and blog posts are generally a pleasure to read precisely because they are clear and balance well explanations with brevity and that is highly non-trivial to do) $\endgroup$
    – Conrad
    Commented Jul 1, 2020 at 14:49
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    $\begingroup$ "Why disturb the gods" , please don't exaggerate ! $\endgroup$
    – Peter
    Commented Jul 5, 2020 at 16:19

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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

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