A Cobham theorem for scalar multiplication
<|reference_start|>A Cobham theorem for scalar multiplication: 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.<|reference_end|>
@article{hieronymi2024a,
    title={A Cobham theorem for scalar multiplication},
    author={Philipp Hieronymi, Sven Manthe, Chris Schulz},
    journal={arXiv preprint arXiv:2407.15118},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.15118},
    primaryClass={math.LO cs.LO}
}