A Cobham theorem for scalar multiplication

Abstract

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

Theorems & Definitions (72)

• proof
• Definition 2.4
• proof
• proof
• proof
• Proposition 2.14
• proof
• proof
• proof : Proof of \ref{['definbounded']}
• Definition 2.17