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\vskip 1cm{\LARGE\bf On Error Sum Functions Formed by \\
\vskip .1in Convergents of Real 
Numbers
}
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\large
Carsten Elsner and Martin Stein\\
Fachhochschule f\"ur die Wirtschaft Hannover\\
Freundallee 15\\
30173 Hannover\\
Germany \\
\href{mailto:Carsten.Elsner@fhdw.de}{\tt Carsten.Elsner@fhdw.de} \\
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\begin{abstract}
Let $p_m/q_m$ denote the $m$-th convergent $(m\geq0)$ from the continued fraction expansion of some real number $\alpha$. We continue our
work on error sum functions defined by
$\mathcal{E}(\alpha) := \sum_{m\geq0} |q_m \alpha - p_m|$ and $\mathcal{E}^*(\alpha) := \sum_{m\geq0} (q_m \alpha - p_m)$
by proving a new density result for the values of $\E$ and $\E^*$. Moreover, we study the function $\mathcal{E}$ with respect to
continuity and compute the integral $\int_0^1 \mathcal{E}(\alpha) \,d\alpha$. We also consider generalized error sum functions for the approximation with algebraic numbers of bounded degrees in the sense of K.Mahler.
\end{abstract}


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