I’ve just uploaded to the arXiv my paper “Almost all Collatz orbits attain almost bounded values“, submitted to the proceedings of the Forum of Mathematics, Pi. In this paper I returned to the topic of the notorious Collatz conjecture (also known as the this blog post. This conjecture can be phrased as follows. Let

Conjecture 1 (Collatz conjecture) One has

Establishing the conjecture for all previous blog post, it is basically at least as difficult as Baker’s theorem, all known proofs of which are quite difficult). However, the situation is more promising if one is willing to settle for results which only hold for “most” of Krasikov and Lagarias that

for all sufficiently large by Terras that for almost all natural density), one has by Allouche to by Korec to cover all

Theorem 2 Let

Thus for instance one has

The difficulty here is one usually only expects to establish “local-in-time” results that control the evolution

However, as observed by Bourgain in the context of nonlinear Schrödinger equations, one can iterate “almost sure local wellposedness” type results (which give local control for almost all initial data from a given distribution) into “almost sure (almost) global wellposedness” type results if one is fortunate enough to draw one’s data from an invariant measure for the dynamics. To illustrate the idea, let us take Korec’s aforementioned result that if

So, one now needs to locate a measure which has better invariance properties under the Collatz dynamics. It turns out to be technically convenient to work with a standard acceleration of the Collatz map known as the Syracuse map

When viewed geometric distribution of mean

respectively. More generally, for any Syracuse random variable, and can be described explicitly as

where

In view of this, any proposed “invariant” (or approximately invariant) measure (or family of measures) for the Syracuse dynamics should take this

for any

A first hint of how to proceed comes from the elementary number theory observation (easily proven by induction) that the rational numbers

are all distinct as

are mostly distinct for “typical” (3) then typically takes a form like (3) of (1) is already enough to get quite a bit of spreading on (2) we have to exploit the mixing effects of the remaining portion of (1) that does not come from (3). After some standard Fourier-analytic manipulations, matters then boil down to obtaining non-trivial decay of the characteristic function of

for any

If the random variable (1) was the sum of independent terms, one could express this characteristic function as something like a Riesz product, which would be straightforward to estimate well. Unfortunately, the terms in (1) are loosely coupled together, and so the characteristic factor does not immediately factor into a Riesz product. However, if one groups adjacent terms in (1) together, one can rewrite it (assuming

where conditional sum of independent random variables. This lets one express the characeteristic function of (1) as an averaged Riesz product. One can use this to establish the bound (4) as long as one can show that the expression

is not close to an integer for a moderately large number (

we have to show that (with overwhelming probability) the random walk

(which we view as a two-dimensional renewal process) contains at least a few points lying outside of

A little bit of elementary number theory and combinatorics allows one to describe the set (4), and thus Theorem 2.