While working in a question about the affine group $\text{Aff}(\mathbb{R})$, I have come up with the following strange question about the real numbers:

Question: Do there exist a non-trivial decomposition $\mathbb{R} = T + T'$ as an additive $\mathbb{Q}$-vector space and another decomposition $\mathbb{R}^{+} = SS'$ as a multiplicative vector space (i.e. $\log(S) + \log(S') = \mathbb{R}$), satisfying the following properties:

1) $S'$ and $T'$ are countable. (Therefore $S$, $T$ are uncountable)

2) $T$ is $S$-invariant, i.e. $ST = T$.

Comment: One observation worth making is that if such a decomposition exist, then $V = \mathbb{R}/T$ is a countable $\mathbb{Q}$-vector space and there is a linear action of $S$ on $V$ because $T$ is $S$ invariant. Now, if $\text{dim}_{\mathbb{Q}}{V} < \infty$, beacuse $S$ is uncountable and the group of linear transformation $GL(V)$ is countable, there is $s\neq1$ in $S$ such that $sv = v$ for all $v \in V$, which implies that $(s-1)V = 0$ and therefore $(s-1)\mathbb{R} \subset T$, which implies that $T = \mathbb{R}$. In conclusion, such non-trivial decomposition can exist only if $\text{dim}_{\mathbb{Q}}(V) = \infty$.

Any information related (or vaguely related) to this is greatly appreciated. Also, I don't know if I have the right tags for this question. Thanks.