Haas–Molnar continued fractions and metric Diophantine approximation

Haas–Molnar maps are a family of maps of the unit interval introduced by A. Haas and D. Molnar. They include the regular continued fraction map and A. Renyi's backward continued fraction map as important special cases. As shown by Haas and Molnar, it is possible to extend the theory of metric diophantine approximation, already well developed for the Gauss continued fraction map, to the class of Haas–Molnar maps. In particular, for a real number $x$, if $(p_n/q_n)_{n\geq 1}$ denotes its sequence of regular continued fraction convergents, set $\theta _n(x)=q_n^2|x- p_n/q_n|$, $n=1,2\dots$. The metric behaviour of the Cesàro averages of the sequence $(\theta _n(x))_{n\geq 1}$ has been studied by a number of authors. Haas and Molnar have extended this study to the analogues of the sequence $(\theta _n(x))_{n\geq 1}$ for the Haas–Molnar family of continued fraction expansions. In this paper we extend the study of $(\theta _{k_n}(x))_{n\geq 1}$ for certain sequences $(k_n)_{n\geq 1}$, initiated by the second named author, to Haas–Molnar maps.