Almost all orbits of the Collatz map attain almost bounded values

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 Colₘᵢₙ(N) := inf n ∈ N Colⁿ(N) denote the minimal element of the Collatz orbit N, Col(N), Col²(N), . The infamous Collatz conjecture asserts that Colₘᵢₙ(N)=1 for all N ∈ N+1 . Previously, it was shown by Korec that for any θ> log 3/log 4 ≈ 0.7924 , one has Colₘᵢₙ(N) ≤ N^θ 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 lim N → ∞ f(N)=+∞ , one has Colₘᵢₙ(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/3ⁿZ 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.