Continued fractions and irrational rotations

Abstract

Let $\alpha \in (0, 1)$ be an irrational number with continued fraction expansion $\alpha =[0; a_1, a_2, \ldots ]$ and let $p_n/q_n= [0; a_1, \ldots , a_n]$ be the nth convergent to $\alpha$. We prove a formula for $p_nq_k-q_np_k$ $(k<n)$ in terms of a Fibonacci type sequence $Q_n$ defined in terms of the $a_n$ and use it to provide an exact formula for $\{n\alpha \}$ for all n.