Geodesic
On error sum functions formed by convergents of real numbers
Journal of integer sequences, Tome 14 (2011) no. 8.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 11J04, 11J70, 11B05, 11B39
Keywords: continued fractions, convergents, approximation of real numbers, error terms, density 
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     author = {Elsner, Carsten and Stein, Martin},
     title = {On error sum functions formed by convergents of real numbers},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {14},
     number = {8},
     year = {2011},
     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/