Small additive groups with $A+ξA=\mathbb R$

We prove that, for every irrational $ξ$, any $F_σ$ additive subgroup of $\R$ has an $F_σ$ $\Q$-vector space extension of the same Hausdorff dimension satisfying $A+ξA=\R$. Iteration gives simultaneous surjectivity for any prescribed countable family of irrational multipliers. Such a space exists in