References & Citations
Mathematics > Probability
Title: Push-forward of geometric distributions under Collatz iteration: Part 1
(Submitted on 18 Apr 2024 (v1), last revised 19 Apr 2024 (this version, v2))
Abstract: Two conjectures are presented. The first, Conjecture 1, is that the pushforward of a geometric distribution on the integers under $n$ Collatz iterates, modulo $2^p$, is usefully close to uniform distribution on the integers modulo $2^p$, if $p/n$ is small enough. Conjecture 2 is that the density is bounded from zero for the incidence of both $0$ and $1$ for the coefficients in the dyadic expansions of $-3^{-\ell }$ on all but an exponentially small set of paths of a geometrically distributed random walk on the two-dimensional array of these coefficients. It is shown that Conjecture 2 implies Conjecture 1. At present, Conjecture 2 is unresolved.
Submission history
From: Mary Rees [view email][v1] Thu, 18 Apr 2024 15:52:43 GMT (22kb)
[v2] Fri, 19 Apr 2024 07:22:31 GMT (22kb)
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