2019/09/08 by Terence Tao, Tao, Terence · 4 voices · 8 citations
#math.PR #math.DS #math.NT
paper · pdf · doi:10.48550/arxiv.1909.03562
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) := infn ∈ 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) ≤ Nθ 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 limN → ∞ 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 an approximate transport 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 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.