Geodesic
On error sum functions formed by convergents of real numbers
Journal of integer sequences, Tome 14 (2011) no. 8
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.
Classification : 11J04, 11J70, 11B05, 11B39
Keywords: continued fractions, convergents, approximation of real numbers, error terms, density 
@article{JIS_2011__14_8_a5,
     author = {Elsner,  Carsten and Stein,  Martin},
     title = {On error sum functions formed by convergents of real numbers},
     journal = {Journal of integer sequences},
     year = {2011},
     volume = {14},
     number = {8},
     zbl = {1255.11036},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2011__14_8_a5/}
}
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Elsner,  Carsten; Stein,  Martin. On error sum functions formed by convergents of real numbers. Journal of integer sequences, Tome 14 (2011) no. 8. http://geodesic.mathdoc.fr/item/JIS_2011__14_8_a5/