Beckman-Quarles type theorems for mappings from R^n to C^n

Abstract

Let G: C^n \times C^n -> C, G((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+ (x_n-y_n)^2. We say that f: R^n -> C^n preserves distance d>0 if for each x,y \in R^n G(x,y)=d^2 implies G(f(x),f(y))=d^2. Let A(n) denote the set of all positive numbers d such that any map f: R^n -> 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 R^n and |x-y|=d then there exists a finite set S(x,y) with {x,y} \subseteq S(x,y) \subseteq R^n such that any map f:S(x,y)->C^n that preserves unit distance preserves also the distance between x and y. We prove: (1) A(n) \subseteq {d>0: d^2 \in Q}, (2) for n>=2 D(n) is a dense subset of (0,\infty). Item (2) implies that each continuous mapping f from R^n to C^n (n>=2) preserving unit distance preserves all distances.