A Cobham theorem for scalar multiplication

Let $\alpha ,\beta \in {\mathbb{R}}_{>0}$ be such that $\alpha ,\beta$ are quadratic and $\mathbb{Q}(\alpha )\mathrm{\ne }\mathbb{Q}(\beta )$. Then every subset of ${\mathbb{R}}^n$ definable in both $(\mathbb{R},<,+,\mathbb{Z},x\mapsto \alpha x)$ and $(\mathbb{R},<,+,\mathbb{Z},x\mapsto \beta x)$ is already definable in $(\mathbb{R},<,+,\mathbb{Z})$. As a consequence we generalize Cobham-Semenov theorems for sets of real numbers to $\beta$-numeration systems, where $\beta$ is a quadratic irrational.