Let $ p_m/q_m$ denote the $ m$-th convergent $ (mgeq0)$ from the continued fraction expansion of some real number $ lpha$. We continue our work on error sum functions defined by $ mathcal{E}(lpha) := sum_{mgeq0} ert q_m lpha - p_mert$ and $ mathcal{E}^*(lpha) := sum_{mgeq0} (q_m lpha - 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}(lpha) ,dlpha$. We also consider generalized error sum functions for the approximation with algebraic numbers of bounded degrees in the sense of Mahler.
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