Haas-Molnar continued fractions and metric Diophantine approximation
For \(x \in [0,1]\), a well-known quantity \(\theta_n(x)\) measures the distance between \(x\) and the \(n\)-th convergent of its continued fraction expansion. \textit{A. Haas} and \textit{D. Molnar} [Trans. Am. Math. Soc. 356, No. 7, 2851--2870 (2004; Zbl 1051.11038)] introduced a family of piecewise Möbius transformations \(T_u:[0,1] \rightarrow [0,1]\) which includes the Gauss map. Those authors studied analogues of \((\theta_n(x))_{n \geq 1}\) for \(T_u\). The paper reviewed here extends results of \textit{R. Nair} [New York J. Math. 3A, 117--124 (1998; Zbl 0894.11032)] from the Gauss map to \(T_u\), studying \((\theta_{k_n}(x))_{n \geq 1}\) for certain sequences \((k_n)_{n \geq 1}\). The authors' results also generalize theorems of \textit{D. Hensley} [New York J. Math. 4, 249--257 (1998; Zbl 0991.11046)] and \textit{A. Haas} and \textit{D. Molnar} [Pac. J. Math. 217, No. 1, 101--114 (2004; Zbl 1080.11060)].
scientific article; zbMATH DE number 6919531
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