Rational approximation with generalised $\alpha $-Lüroth expansions
Volume 215 / 2024
Abstract
For a fixed partition $\alpha $, each real number $x \in (0,1)$ can be represented by many different generalised $\alpha $-Lüroth expansions. Each such expansion produces a sequence $(p_n/q_n)_{n \ge 1}$ of rational approximations to $x$. We study the corresponding approximation coefficients $(\theta_n(x))_{n \ge 1}$, which are given by $$ \theta _n (x): = q_n \biggl|x-\frac{p_n}{q_n}\biggr|.$$ We give the cumulative distribution function and the expected average value of the $\theta _n$, and we identify which generalised $\alpha $-Lüroth expansion has 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 [J. Barrionuevo et al., Acta Arith. 74 (1996), 311–327].
