Rational approximation with generalised \(\alpha\)-Lüroth expansions (Q6619740)

In this paper, generalised \(\alpha\)-Lüroth expansions are considered, of which Lüroth expansions and alternating Lüroth expansions are specific examples. These generalised \(\alpha\)-Lüroth expansions are defined by describing the dynamical algorithms for obtaining them. Define the approximation coefficients \((\theta_n^\varepsilon(x))_{n\ge 1}\) by \(\theta_n^\varepsilon(x):=q_n|x-p_n/q_n|\), where \((p_n/q_n)_{n\ge 1}\) are the rational approximations to \(x\in(0,1)\). In this paper, by generalizing the result in [\textit{J. Barrionuevo} et al., Acta Arith. 74, No. 4, 311--327 (1996; Zbl 0848.11039)] to arbitrary partitions, it is proved that for any class of Lüroth partition \(\alpha\), \(\varepsilon\in\{0,1\}^{\mathbb N}\), \(z\in(0,1]\) and Lebesgue a.e. \(x\in[0,1]\) the limit \N\[\N\lim_{N\to\infty}\frac{\#\{1\le n\le N:\theta_n^\varepsilon(x)<z\}}{N}\N\]\Nexists. The authors also analyse the structure of the set \(\mathcal M\alpha\) of possible values that \(M_\varepsilon:=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\theta_i^\varepsilon(x)\) can take, by answering a question in [loc. cit.].