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\begin{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 \(\sum_{m}
\varepsilon_m \), \(\sum_{m} |\varepsilon_m| \), 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.
\end{abstract}

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