Exercise 10, § 8

Let $(d_n)$ be a base sequence, let $a_n = d_n / d_{n-1}$ ($n \geq 1$), let $(n_i)$ be a strictly increasing sequence of integers, and let $(b_i)$ be a sequence of integers such that $0 < b_i < a_{n_i} - 1$ for each $i$. Let $G$ be the set of all $x \in [0, 1]$ such that, if $\sum_{n=1}^{\infty} u_n / d_n$ is the expansion of $x$, or the improper expansion if it exists, then $u_{n_i} \neq b_i$ for all $i \in \mathbf{N}$. Show that $G$ is a totally disconnected perfect set, and that if $l$ is the sum of the lengths of the intervals contiguous to $G$ and contained in $[0, 1]$, then