About this video
This talk was a joint Dynamical Systems and Stochastics seminar at the University of Liverpool.
Abstract: The Collatz map sends an integer n to n/2 if n is even and 3n+1 if n is odd. So the orbit of 1 is 1,4,2,1,4,2,1..
The orbit of 3 is 3,10,5,16,8,4,2,1.. The famously intractable Collatz conjecture, which is thought to have been first circulated by word of mouth in 1950, states that the orbit of every strictly positive integer ends in the cycle 1,4,2..
It is not even known if this problem is decidable. John Conway proved in the 1950's that a more general problem is not decidable. What positive information we have about the Collatz map is largely, although not entirely, of a probabilistic nature. It was shown in the 1970's that, for most positive integers n, in the sense of positive density, the Collatz orbit of n passes below n. Terence Tao's 2019 paper ``Almost all Collatz orbits attain almost bounded values'' is a powerful strengthening of this basic result.
Most work on the Collatz conjecture is focussed on ``typical'' integers, which essentially means those n such that the orbit of n starts by passing below n in a reasonable amount of time. Tao's method shows that the orbits such numbers typically continue to decrease for some time.
This talk will concentrate on numbers which are not necessarily typical -- by looking not at the standard measures of uniform density on intervals of integers, but at other measures on integers: the so called geometric measures, and integers which are typical for such a measure. These are numbers whose Collatz orbits start by increasing, depending on the mean of the geometric measure. A chain of conjectures will be stated which, if true, show that the Collatz orbits of these sparser numbers do later decrease for some time. A first weaker version of the final conjecture in the chain has been proved: not enough to be the basis of an induction, but a step in the right direction.
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