You need to know: Set of positive integers, logarithm, limits.
Background: The Collatz map is defined by (i) when is odd and (ii) when n is even. For any , let denote the k-th iterate of let be the Collatz orbit, and let denote the minimal element of the Collatz orbit .
We say that subset has logarithmic density 1, if . We say that a property P holds for almost all (in the sense of logarithmic density) if the subset of for which P holds has logarithmic density 1.
The Theorem: On 8th September 2019, Terence Tao submitted to arxiv a paper in which he proved that for any function such that , one has for almost all (in the sense of logarithmic density).
Short context: The famous Collatz conjecture predicts that for all , but it remains well beyond reach of the current methods. As a partial progress, Terras proved in 1976 that for almost all n. In 1979, Allouche improved this to for almost all n, for any fixed constant . In 1994, Korec proved the same result for any . The Theorem implies that one can take any , and, more generally, for almost all n and any function f going to infinity with n. For example, one can take .
Links: Free arxiv version of the original paper is here.