A [new preprint by Terry Tao](https://arxiv.org/abs/1909.03562) 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:



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> 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 $1-\delta$.
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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.



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> Are there any values of $\delta<1$ for which an *explicit* upper bound for $C\_\delta$ is known?
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The second essentially asks if anything sort-of-explicit can be deduced from Tao's result.



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> Is the function $N\mapsto C\_{1/N}$ upper bounded by a computable function?
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(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)



Explicit bounds from Tao's result on Collatz conjecture
In v2 of the arXiv preprint, there is more detail, including fairly explicit dependence on $\delta$.


