A Cobham theorem for scalar multiplication

Abstract

Let $\hat{I}\pm ,{\hat{I}}^2\in {\mathbb{R}}_{>0}$ be such that $\hat{I}\pm ,{\hat{I}}^2$ are quadratic and $\mathbb{Q}(\hat{I}\pm )\ne \mathbb{Q}({\hat{I}}^2)$. Then every subset of ${\mathbb{R}}^n$ definable in both $(\mathbb{R},<,+,\mathbb{Z},x\mapsto \hat{I}\pm x)$ and $(\mathbb{R},<,+,\mathbb{Z},x\mapsto {\hat{I}}^{2}x)$ is already definable in $(\mathbb{R},<,+,\mathbb{Z})$. As a consequence we generalize Cobham-Semenov theorems for sets of real numbers to ${\hat{I}}^2$-numeration systems, where ${\hat{I}}^2$ is a quadratic irrational.