On mappings of $\mathbb{Q}^d$ to $\mathbb{Q}^d$ that preserve distances 1 and $\sqrt 2$ and the Beckman-Quarles Theorem

Abstract.

Benz proved that every mapping $f:\mathbb{Q}^d \to \mathbb{Q}^d$ that preserves the distances 1 and 2 is an isometry, provided d ≥ 5. We prove that every mapping $f:\mathbb{Q}^d \to \mathbb{Q}^d$ that preserves the distances 1 and $\sqrt 2$ is an isometry, provided d ≥ 5.