On the Distribution of the Digits in Lüroth Expansions

Abstract

For x ∈ [0, 1), let [d₁(x), d₂(x), . . .] be its Lüroth expansion, and let {p_n(x)/qn(x)}_n≥1 be the sequence of convergents of x. In this paper, we prove that the Hausdorff dimension of the exceptional set

$$ {F}_{\alpha}^{\beta }=\left\{x\in \left[\left.0,1\right)\right.:\underset{n\to \infty }{\lim}\operatorname{inf}\frac{\log\ {d}_{n+1}(x)}{-\log \left|x-\frac{p_n(x)}{q_n(x)}\right|}=\alpha, \underset{n\to \infty }{\lim}\sup \frac{\log\ {d}_{n+1}(x)}{-\log \left|x-\frac{p_n(x)}{q_n(x)}\right|}\ge \beta \right\} $$

is (1 − β)/2 or 1 − β according to α > 0 or α = 0. This extends an earlier result of Tan and Zhang.