The almost PV behavior of some far from PV algebraic integers (Q1343252)

For a quadratic irrationality \(\vartheta\) of discriminant \(\Delta\) the authors study the behaviour of the sequence \(a_ n\) defined inductively by \(a_ 1 =1\), \(a_{n+1}= sa_ n+ t[\vartheta a_ n]\), with \(s\), \(t\) being given rational integers. They show that if \(\vartheta\) is a quadratic unit of negative norm and odd positive trace then for infinitely many choices of \(s\) and \(t\) the highest power of \(\Delta\) dividing \(a_ n\) tends to infinity. For other quadratic integers \(\vartheta>1\) they prove the existence of infinitely many choices for \(s\) and \(t\) so that the elements of the sequence \(a_ n\) lie for \(n\geq 3\) in at most two residue classes \(\text{mod } \Delta\).