We discuss the following general question and some of its extensions. Let (εk)k≥1 be a sequence with values in {0,1}, which is not ultimately periodic. Define ξ:=∑ k≥1εk/2k and ξ′:=∑ k≥1εk/3k. Let 𝒫 be a property valid for almost all real numbers. Is it true that at least one among ξ and ξ′ satisfies 𝒫?

We show that for a given base $b$ and a proper subset $E\subset\{0,\dots,b-1\}$, $\#E\ltb-1$, the set of numbers $x\in[0,1]$ that have no digits from $E$ in their expansion to base $b$ consists almost exclusively of $S^*$-numbers of type at most $\min\{2,\log b/\log(b-\#E)\}$. We also give upper bounds on the Hausdorff dimension of some exceptional sets.

We divide infinite sequences of subword complexity 2n+1 into four subclasses with respect to left and right special elements and examine the structure of the subclasses with the help of Rauzy graphs. Let k ≥ 2 be an integer. If the expansion in base k of a number is an Arnoux-Rauzy word, then it belongs to Subclass I and the number is known to be transcendental. We prove the transcendence of numbers with expansions in the subclasses II and III.