The almost PV behavior of some far from PV algebraic integers Original Research Article

This paper studies divisibility properties of sequences defined inductively by a1 = 1, an+1 =san + t⌊θam⌋, where s,t are integers, and θ is a quadratic irrationality. Under appropriate hypotheses (especially that s + tθ be a PV-number) it is proved that the highest power of Δ that divides an, where Δ is the discriminant of θ, tends to infinity. This is noteworthy in that truncation would normally be expected to destroy any simple algebraic structure. Moreover, we establish related results that imply the an are not uniformly distributed modulo Δ in cases where the smaller conjugate of s + tθ exceeds 1 in modulus (the non-PV case).