Small additive groups with $A+\xi A=\mathbb{R}$

We prove that, for every irrational $\xi$, any $F_{\sigma }$ additive subgroup of $\mathbb{R}$ has an $F_{\sigma }$ $\text{Q}$-vector space extension of the same Hausdorff dimension satisfying $A+\xi A=\mathbb{R}$. Iteration gives simultaneous surjectivity for any prescribed countable family of irrational multipliers. Such a space exists in every prescribed dimension $d\in [0,1)$. We also construct in ZFC a zero-dimensional lightface ${\mathrm{\Delta }}_3^0$ $\text{Q}$-vector subspace $A$ for which $A+\xi A=\mathbb{R}$ holds for every irrational $\xi$, answering both parts of Question~2 of Ye, Yu, and Zhao. This universal group is a difference of two $F_{\sigma }$ sets, but is neither $F_{\sigma }$ nor $G_{\delta }$. Both constructions use direct inductions on finite binary strings and admit effective versions. As applications, we obtain zero-dimensional $F_{\sigma }$ groups whose Cartesian squares have Hausdorff dimension one, and relate their dilate sums to Marstrand’s projection theorem.