Uniform distribution of two-term recurrence sequences

Abstract:

Let ${u_0}, {u_1}, A, B$ be rational integers and for $n \geqslant 2$ define ${u_n} = A{u_{n - 1}} + B{u_{n - 2}}$. The sequence $({u_n})$ is clearly periodic modulo $m$ and we say that $({u_n})$ is uniformly distributed modulo $m$ if for every $s$, every residue modulo $m$ occurs the same number of times in the sequence of residues ${u_s}, {u_{s + 1}}, \ldots , {u_{s + N - 1}}$, 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 ${p^k}f$, then every residue modulo ${p^k}$ appears exactly once in the sequence ${u_s}, {u_{s + f}}, \ldots , {u_{s + ({p^k} - 1)f}}$, for every $s$. We also characterize those composite $m$ for which this more stringent property holds.