Rational approximation with generalised $\alpha$-L\"{u}roth expansions

For a fixed $\alpha$, each real number $x \in (0,1)$ can be represented by many different generalised $\alpha$-L\"uroth expansions. Each such expansion produces for the number $x$ a sequence of rational approximations $(\frac{p_n}{q_n})_{n \ge 1}$. In this paper we study the corresponding approximation coefficients $(\theta_n(x))_{n \ge 1}$, which are given by $$\theta_n (x): = q_n \left|x-\frac{p_n}{q_n}\right|.$$ We give the cumulative distribution function and the expected average value of the $\theta_n$ and we identify which generalised $\alpha$-L\"uroth expansion gives the best approximation properties. We also analyse the structure of the set $\mathcal M_\alpha$ of possible values that the expected average value of $\theta_n$ can take, thus answering a question from \cite{Barrionuevo-Burton-Dajani-Kraaikamp-1994}.