Chains of Cantor subspaces of Mahler $T$-Numbers and the middle-third Cantor set

We investigate Mahler's real $T$-numbers through Cantor-type constructions inside $\mathcal{T}\cap \mathbb{G}$, where $\mathbb{G}$ is the middle-third Cantor set and $\mathcal{T}$ is the set of real Mahler $T$-numbers. We build explicit families ${\mathcal{T}}^{(t)}$ which are homeomorphic to Cantor space and use them to analyze structural, combinatorial, and additive properties of $\mathcal{T}$. Our results include the existence of descending chains of Cantor subsets of $\mathcal{T}$ of length $\mathfrak{c}$; a characterization of ternary expansions in ${\mathcal{T}}^{(t)}$, showing non-normality but maximal block complexity under sparse forcing, linked to the Adamczewski–Bugeaud criterion; and a sumset theorem proving that for suitable parameters $t_1,t_2$ one has the interval identity ${\mathcal{T}}^{(t_1)}+{\mathcal{T}}^{(t_2)}=[0,1]$, which yields the global corollary $\mathcal{T}+\mathcal{T}=\mathbb{R}$ (Erdős property) by integer translation invariance. We also discuss implications for cardinal invariants and entropy of the shift map, highlighting the interplay between thin Diophantine sets and large additive structure. To address a natural concern about existence, we include a non-emptiness lemma which shows that our scheduled deletion-and-witness procedure always leaves a non-empty perfect set ${\mathcal{T}}^{(t)}$.