The Beckman-Quarles theorem for continuous mappings from R² to C²

الأصل · EN

Let ϕ((x₁,x₂),(y₁,y₂))=(x₁-y₁)²+(x₂-y₂)². We say that f:R² -> C² preserves distance d>=0 if for each x,y ∈ R² ϕ(x,y)=d² implies ϕ(f(x),f(y))=d². We prove that if x,y ∈ R² and |x-y|=(2√2/3)ᵏ · (√3)ˡ (k,l are non-negative integers) then there exists a finite set x,y S(x,y) R² such that each unit-distance preserving mapping from S(x,y) to C² preserves the distance between x and y. It implies that each continuous map from R² to C² preserving unit distance preserves all distances.