A Cobham theorem for scalar multiplication

TL;DR

This paper proves a Cobham-type theorem for scalar multiplication by quadratic irrationals, showing that sets definable in two different such systems are already definable in the base structure, extending Cobham-Semenov results.

Contribution

It generalizes Cobham's theorem to scalar multiplication by quadratic irrationals, linking definability across different numeration systems.

Findings

Sets definable in two distinct quadratic scalar systems are already definable in the base structure.

Generalization of Cobham-Semenov theorems to quadratic irrational bases.

Establishes conditions under which definability is preserved across different numeration systems.

Abstract

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.