On the combinatorial structure of \(3N+1\) predecessor sets
The ``Collatz''-problem (or ``\(3N+ 1\)''- or ``Hasse''- or ``Syracuse''- or ``Kakutani''-problem) is to prove that for every \(n\in \mathbb{N}\) there exists a \(k\) with \(T^{(k)}(n)= 1\) where the function \(T(n)\) takes odd numbers \(n\) to \((3n+ 1)/2\) and even numbers \(n\) to \(n/2\). In the note under review, the author studies not as usual the trajectory \((T^{(k)}(a))_{k\in \mathbb{N}_0}\) for \(a\in \mathbb{N}\), but its predecessor set which consists of all integers \(n\) whose iterates \(T^{(k)}(n)\) eventually hit \(a\). He gives a representation of these sets and studies their combinatorial structure. It turns out that the well-known coefficient stopping time conjecture in the original ``\(3N+ 1\)''-problem could be formulated in terms of the predecessor sets. In the last section, the author shows how density considerations are related to the asymptotic behaviour at the origin of a certain smooth function on the unit interval which would also convey new information on the original ``\(3N+ 1\)''-problem.
scientific article; zbMATH DE number 858089