Rational approximation with generalised $\alpha$-Lüroth expansions

Abstract

For a fixed $\alpha$, each real number $x\in (0,1)$ can be represented by many different generalised $\alpha$-Lüroth 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 \[ θ_n (x): = q_n \left|x-\frac{p_n}{q_n}\right|.\] We give the cumulative distribution function and the expected average value of the ${\theta }_n$ and we identify which generalised $\alpha$-Lüroth 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}.

23 pages,3 figures