Summary: 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 Mahler.

11J04, 11J70, 11B05, 11B39

continued fractions, convergents, approximation of real numbers, error terms, density