Almost all Collatz Orbits Attain Almost Bounded Values - Terence Tao скачать в хорошем качестве

Almost all Collatz Orbits Attain Almost Bounded Values - Terence Tao

Members' Colloquium Topic: Almost all Collatz Orbits Attain Almost Bounded Values Speaker: Terence Tao Affiliation: University of California, Los Angeles; Member, School of Mathematics Date: March 13, 2023 Define the Collatz map Col on the natural numbers by setting Col(n) to equal 3n+1 when n is odd and n/2 when n is even. The notorious Collatz conjecture asserts that all orbits of this map eventually attain the value 1. This remains open, even if one is willing to work with almost all orbits rather than all orbits. We show that almost all orbits n, Col(n), Col^2(n), ... eventually attain a value less than f(n), for any function f that goes to infinity (no matter how slowly). A key step is to obtain an approximately invariant (or more precisely, self-similar) measure for the (accelerated) Collatz dynamics.