Beckman-Quarles type theorems for mappings from $\mathbb{R}^n$ to $\mathbb{C}^n$

Summary.

Let $\varphi_{n} : \mathbb{C}^{n} \times \mathbb{C}^{n} \to \mathbb{C}, \varphi_{n}((x_{1}, \dots, x_{n}), (y_{1}, \dots, y_{n})) = (x_{1} - y_{1})^{2} + \dots + (x_{n} - y_{n})^{2}.$ We say that $f : \mathbb{R}^{n} \to \mathbb{C}^{n}$ preserves the distance d ≥ 0 if for each $x,y \in \mathbb{R}^n \, \varphi_{n}(x,y) = d^{2}$ implies $\varphi_{n}(f(x), f(y)) = d^{2}.$ Let A _n denote the set of all positive numbers d such that any map $f : \mathbb{R}^{n} \to \mathbb{C}^{n}$ that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if $x,y \in \mathbb{R}^n$ and $|x - y| = d$ then there exists a finite set S _xy with $\{x,y\} \subseteq S_{xy} \subseteq \mathbb{R}^{n}$ such that any map $f : S_{xy} \to \mathbb{C}^n$ that preserves unit distance preserves also the distance between x and y. Obviously, $D_{n} \subseteq A_{n}.$ We prove: (1) $A_{n} \subseteq \{d > 0 : d^{2} \in \mathbb{Q}\},$ (2) for n ≥ 2 D _n is a dense subset of $(0, \infty).$ (2) implies that each mapping f from $\mathbb{R}^n$ to $\mathbb{C}^n$ (n ≥ 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on $\mathbb{R}^n$ and $\mathbb{C}^n.$