Uniform distribution of two-term recurrence sequences

Let (FORMULA PRESENTED), A, B be rational integers and for (FORMULA PRESENTED). The sequence (un) is clearly periodic modulo m and we say that (un) is uniformly distributed modulo m if for every 5, every residue modulo m occurs the same number of times in the sequence of residues (FORMULA PRESENTED), where N is the period of (u_n) modulo m. If (u_n) is uniformly distributed modulo m then m divides N, so we write N=mf. Several authors have characterized those m for which (u_n) is uniformly distributed modulo m. In fact in this paper we will show that a much stronger property holds when m = p^k, p a prime. Namely, if (u_n) is uniformly distributed modulo p^k with period p^kf, then every residue modulo p^k appears exactly once in the sequence (FORMULA PRESENTED), for every s. We also characterize those composite m for which this more stringent property holds.