On the Value Distribution of Error Sums for Approximations with Rational Numbers

Let α be a real number with convergents pm/qm from the continued fraction expansion of α. In this paper we investigate the functions E(α) := � m≥0 |αqm − pm| and E∗(α) := � m≥0(αqm − pm) depending only on α and prove that they take every value in [0, (1 + √ 5)/2] and [0, 1], respectively. For any sequence (αμ)μ≥1, which is uniformly distributed modulo 1, we show that both sequences (E(αμ))μ≥1 and (E(αμ))μ≥1 are not uniformly distributed. Among other things the proofs rely on an inequality for the function E(α), which improves a former result of the first named author.