Beckman-Quarles type theorems for mappings from Rn to Cn
Abstract
Let G: Cn × Cn -> C, G((x1,...,xn),(y1,...,yn))=(x1-y1)2+...+ (xn-yn)2. We say that f: Rn -> Cn preserves distance d>0 if for each x,y ∈ Rn G(x,y)=d2 implies G(f(x),f(y))=d2. Let A(n) denote the set of all positive numbers d such that any map f: Rn -> Cn 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 ∈ Rn and |x-y|=d then there exists a finite set S(x,y) with x,y ⊂eq S(x,y) ⊂eq Rn such that any map f:S(x,y)->Cn that preserves unit distance preserves also the distance between x and y. We prove: (1) A(n) ⊂eq d>0: d2 ∈ Q, (2) for n>=2 D(n) is a dense subset of (0,∞). Item (2) implies that each continuous mapping f from Rn to Cn (n>=2) preserving unit distance preserves all distances.
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