On a generalization of the \(3x+1\) problem (Q5917728)

The Collatz-problem (or \(3x + 1-\) 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 a generalization of the following form: Let \(\beta\) be any real number \(> 1\) and define on \(\mathbb{Z}\) the function \(T_\beta\) which takes odd numbers \(n\) to \(\lceil \beta n \rceil\) and even numbers \(n\) to \(n/2\). Hence \(T_{3/2} = T\). Although it is somewhat unusual to formulate a conjecture when it is clear that there are many failing numbers the author gives the following definition: \(C_\beta\) is true iff the set of periodic points \[ L_\beta : = \bigl\{ n \in \mathbb{N} \mid T_\beta^{(k)} (n) = n \text{ for some } k \geq 1 \bigr\} \] is finite and for any integer \(n\), there exists some iterate \(T_\beta^{(k)} (n) \in L_\beta\). In the case \(\beta = 3/2\) with \(L_{3/2} = \{1,2\}\), the conjecture \(C_{3/2}\) is equivalent with the \(3x + 1\) conjecture. Generalizing the proof technique by \textit{R. Terras} [Acta Arith. 30, 241- 252 (1976; Zbl 0348.10037)] it is proved that for any transcendental number \(\beta\) with \(1 < \beta < 2\) as well as for any rational number with an even denomiantor (in lowest terms) in that interval the sequence \((T_\beta^{(k)} (n))_{k \in \mathbb{N}}\) decreases for almost all natural numbers \(n\). If this happens for all sufficiently large numbers \(n\), \(C_\beta\) is true. On the other hand, \textit{S. Brocco} [J. Number Theory 52, 173-178 (1995; Zbl 0820.11007)] showed that \(C_\beta\) is false for all Pisot numbers \(> 1\). For the golden number \(\beta = (\sqrt 5 + 1)/2\) the author gives an elementary proof. If \(\beta > 2\) and \(\beta \in U\), where \(U\) denotes the uniform distribution set (too complicated to give the exact definition here), then almost all iterates of \(T_\beta\) tend to infinity. So the interesting problem to characterize the set of numbers \(\beta\) for which \(C_\beta\) is true remains.