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

Abstract

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 every prescribed dimension $d\in[0,1)$. We also construct in ZFC a zero-dimensional lightface $Δ^0_3$ $\Q$-vector subspace $A$ for which $A+ξA=\R$ holds for every irrational $ξ$, answering both parts of Question~2 of Ye, Yu, and Zhao. This universal group is a difference of two $F_σ$ sets, but is neither $F_σ$ nor $G_δ$. Both constructions use direct inductions on finite binary strings and admit effective versions. As applications, we obtain zero-dimensional $F_σ$ groups whose Cartesian squares have Hausdorff dimension one, and relate their dilate sums to Marstrand's projection theorem.