Exercise 10, § 8

Let $(d_n)$ be a basis sequence, $a_n = d_n/d_{n-1}$ ($n \geq 1$), $(n_i)$ a strictly increasing sequence of integers, $(b_i)$ a sequence of integers such that $0 < b_i < a_{n_i} - 1$ for every $i$. Let $G$ be the set of the $x \in (0, 1)$ such that for the development $\sum_{n=1}^\infty u_n/d_n$ of $x$, or the improper development of $x$ when there is one, one has $u_{n_i} \neq b_i$ for every $i \in \mathbf{N}$. Now prove that $G$ is a perfect totally discontinuous set, and that if $l$ is the sum of the lengths of the intervals contiguous to $G$, contained in $(0, 1)$, one has $1 - l = \prod_{i=0}^\infty \left(1 - \frac{1}{a_{n_i}}\right)$.