It is not well-known that for each separable real locally convex space, its Bohr compactification is isomorphic as a topological group to the Bohr compactification of the topological group $\mathbb{R}$ of all real numbers. This is the case for each separable real Banach space. In this expository research note, we state the results explicitly and provide accessible proofs.

During a first or second course in number theory, students soon encounter several sets of "number theoretic interest". These include basic sets such as the rational numbers, algebraic numbers, transcendental numbers, and Liouville numbers, as well as more exotic sets such as the constructible numbers, normal numbers, computable numbers, badly approximable numbers, the Mahler sets S, T and U, and sets of irrationality exponent $m$, among others. Those exposed to some measure theory soon make a curious observation regarding a common property seemingly shared by all these sets: each of the sets has Lebesgue measure equal to zero, or its complement has Lebesgue measure equal to zero. In this expository note, we explain this phenomenon.

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 ${\mathcal{T}}^{(t)}$.