TITLE: Explicit bounds from Tao's result on Collatz conjecture
QUESTION [12 upvotes]: A new preprint by Terry Tao has recently appeared and has established some interesting results regarding the topic of Collatz conjecture. I will not cite the precise result, but rather an equivalent formulation which Tao notes in his Remark 1.4:

For any $\delta>0$ there exists a constant $C_\delta$ such that $\mathrm{Col_{min}}(N)\leq C_\delta$ for all $N$ in a subset of $\mathbb N+1$ of logarithmic density at least $1-\delta$.

My question concerns whether anything is known about the rate of growth of $C_\delta$ as $\delta\to 0$. I will pose two specific questions in that regard. The first one I imagine might have been known before Tao's recent result.

Are there any values of $\delta<1$ for which an explicit upper bound for $C_\delta$ is known?

The second essentially asks if anything sort-of-explicit can be deduced from Tao's result.

Is the function $N\mapsto C_{1/N}$ upper bounded by a computable function?

(Note that I don't think the answer is "obviously yes" from mere existence of $C_\delta$, since checking that a particular value works seems to be $\Sigma^0_3$ (there exists $N_0$ such that for all $N>N_0$ there exists $M$ such that in $M$ steps at least $1-\delta$ of numbers below $N$ go below $C_\delta$) and is not obviously any lower)

REPLY [2 votes]: In v2 of the arXiv preprint, there is more detail, including fairly explicit dependence on $\delta$.