Collatz Attractors Are Space-Filling

Abstract

The algebraic topology of Collatz attractors (or “Collatz Feathers”) remains very poorly understood. In particular, when pushed to infinity, is their set of branches discrete or continuous? Here, we introduce a fundamental theorem proving that the latter is true. For any odd x, we first define Axn as the set of all odd numbers with Syr(x) in their Collatz orbit and up to n more digits than x in base 2. We then prove limn→∞|Axn|2n+c≥1 with c>−4 for all x and, in particular, c=0 for x=1, which is a result strictly stronger than that of Tao 2019.