See https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/ https://terrytao.wordpress.com/2019/09/10/almost-all-colla

See https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/ https://terrytao.wordpress.com/2019/09/10/almost-all-collatz... for a more detailed explanation. His actual result is this. Suppose that f(N) is a function from the natural numbers to the reals that has infinity as a limit. Let X be the set of natural numbers who eventually go below f(N). Then the logarithmic density of X is 1. Here logarithmic density is the limit of the following ratio: sum(1/n for n in X and < N) / sum(1/n for n < N) Now you just have to pick f that grows very, very slowly. For example f(N) = log(log(log(x)))).