Let $\alpha,\beta \in \mathbb{R}_{>0}$ be such that $\alpha,\beta$ are quadratic and $\mathbb{Q}(\alpha)\neq \mathbb{Q}(\beta)$. 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.

翻译：设 $\alpha,\beta \in \mathbb{R}_{>0}$ 为二次实数，且满足 $\mathbb{Q}(\alpha)\neq \mathbb{Q}(\beta)$。则任何在结构 $(\mathbb{R},{<},+,\mathbb{Z},x\mapsto \alpha x)$ 与 $(\mathbb{R},{<},+,\mathbb{Z},x\mapsto \beta x)$ 中均可定义的 $\mathbb{R}^n$ 子集，必然已在结构 $(\mathbb{R},{<},+,\mathbb{Z})$ 中可定义。作为推论，我们将关于实数集的Cobham-Semenov定理推广至以二次无理数 $\beta$ 为基的 $\beta$-进位制系统。