A note on Mignosi's generalization of the \((3X+1)\)-problem (Q1893467)

Let \(\alpha,r\) be any real numbers with \(1<\alpha <2\) and define the function \(T_{\alpha, r}(n)\) for \(n\in \mathbb{N}\) by \([\alpha n+ r]\) if \(n\) is odd and by \({n\over 2}\) otherwise. The set of purely periodic points of \(T_{\alpha,r}\) under iteration is denoted \(L_{\alpha,r}\). The still open ``Collatz''-problem (or ``\(3x+1\)''- or ``Hasse''- or ``Syracuse''- or ``Kakutani''-problem) now reads as \(L_{3/2,1/2}= \{1, 2\}\) and for every \(n\in \mathbb{N}\) there is a \(k\in \mathbb{N}\), such that \(T^{(k)}_{3/2,1/2}(n)\in L_{3/2,1/2}\). For \((\alpha,r)= (\sqrt{2},1)\) and in some other cases F. Mignosi proved the following: Generalized \((3x+1)\)-conjecture. The set \(L_{\alpha,r}\) is always finite, and for every integer \(n\in \mathbb{Z}\) there is a \(k\in \mathbb{N}\), such that \(T^{(k)}_{\alpha,r}(n)\in L_{\alpha,r}\). In the note under review the author shows that this conjecture fails for \(n\) in a set of positive density \(S\) if \(\alpha\) is a Salem or a PV number, with some restriction on the pair \((\alpha,r)\) if \(\alpha> {3\over 2}\), more precisely, if \(n\in S\) then \(T_{\alpha,r}(n)\in S\) and \(T_{\alpha,r}(n)> n\).