On the Behavior of Unbounded Collatz Sequences

Abstract

The aim of this paper is to show a peculiar behavior of a (hypothetical) Collatz sequence going to infinity. We study the associated Syracusa sequence (the odd elements of the former) and show that the limit set of a conveniently normalized sequence is the whole unit interval. In particular, for any positive integer there is a subsequence whose elements' expansions in base 3 begin (from the left) with the expansion of the given number.