Abstract:

Let \(p_n/q_n \) be the \(n\)-th convergent of a real irrational number \(\alpha \), and let \(\varepsilon_n = \alpha q_n-p_n \). In this paper we investigate various sums of the type \(\sum_{m} \varepsilon_m \), \(\sum_{m} \vert\varepsilon_m\vert \), and \(\sum_{m} \varepsilon_m x^m \). The main subject of the paper is bounds for these sums. In particular, we investigate the behaviour of such sums when \(\alpha \) is a quadratic surd. The most significant properties of the error sums depend essentially on Fibonacci numbers or on related numbers.


Jeffrey Shallit 2011-01-28