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On Error Sum Functions Formed by
Convergents of Real Numbers Carsten Elsner and Martin Stein
Fachhochschule für die Wirtschaft Hannover
Freundallee 15
30173 Hannover
Germany
mailto:Carsten.Elsner@fhdw.deCarsten.Elsner@fhdw.de

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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 $\mathcal{E}(\alpha) := \sum_{m\geq0} \vert q_m \alpha - p_m\vert$ and $\mathcal{E}^*(\alpha) := \sum_{m\geq0} (q_m \alpha - p_m)$ by proving a new density result for the values of $\mathcal{E}$ and $\mathcal{E}^*$. 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.