Rational approximation with generalised -Lüroth expansions
Tom 215 / 2024
Streszczenie
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].
