On Error Sum Functions Formed by Convergents of Real Numbers

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

                Carsten.Elsner@fhdw.de

Abstract

Let p_{m}/q_{m} denote the m-th convergent (m ≥ 0) from the continued fraction expansion of some real number α. We continue our work on error sum functions defined by E(α) := ∑_{m≥0} |q_{m}α − p_{m}| and E^{∗}(α) := ∑_{m≥0}(q_{m}α − p_{m}) by proving a new density result for the values of E and E^{∗}. Moreover, we study the function E with respect to continuity and compute the integral ∫_{0}^{1} E(α) dα. We also consider generalized error sum functions for the approximation with algebraic numbers of bounded degrees in the sense of K.Mahler.