Journal of Integer Sequences, Vol. 14 (2011), Article 11.8.6

On Error Sum Functions Formed by Convergents of Real Numbers

Carsten Elsner and Martin Stein
Fachhochschule für die Wirtschaft Hannover
Freundallee 15
30173 Hannover

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 Mahler.

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(Concerned with sequences A000045 A007676 A007677 A041008 A041009.)

Received May 13 2011; revised versions received July 12 2011; August 17 2011. Published in Journal of Integer Sequences, September 25 2011.

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