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[News] Progress on Collatz Conjecture

Post icon  Posted 10 September 2019 - 05:58 PM

Terry Tao has made huge progress on the Collatz Conjecture!



Define the \emph{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∈NColn(N) denote the minimal element of the Collatz orbit N,Col(N),Col2(N),. The infamous \emph{Collatz conjecture} asserts that Colmin(N)=1 for all N∈N+1. Previously, it was shown by Korec that for any θ>log3log4≈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 \emph{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.

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