Almost all orbits of the Collatz map attain almost bounded values

Abstract Define the Collatz map {Col} N+1 N+1 on the positive integers N+1 = \1, 2, 3, \ by setting {Col} (N) equal to 3N+1 when N is odd and N/2 when N is even, and let {Col} (N): = ₍ ₍ {Col}ⁿ (N) denote the minimal element of the Collatz orbit N, {Col} (N), {Col}² (N),. The infamous Collatz conjecture asserts that {Col} (N) =1 for all N N+1. Previously, it was shown by Korec that for any > 3 4 0. 7924, one has {Col} (N) N^ for almost all N N+1 (in the sense of natural density). In this paper, we show that for any function f N+1 R with ₍ f (N) =+, one has {Col} (N) f (N) for almost all N N+1 (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a 3 -adic cyclic group Z/3ⁿ Z at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.