A discrete form of the Beckman-Quarles theorem for mappings from R2 (C2) to F2, where F is a subfield of a commutative field extending R (C)
Abstract
Let F be a subfield of a commutative field extending R. Let phin:Fn × Fn ->F, phin((x1,...,xn),(y1,...,yn))=(x1-y1)2+...+(xn-yn)2. We say that f:Rn->Fn preserves distance d>=0 if for each x,y ∈ Rn |x-y|=d implies phin(f(x),f(y))=d2. Let An(F) denote the set of all positive numbers d such that any map f:Rn->Fn that preserves unit distance preserves also distance d. Let Dn(F) 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)->Fn that preserves unit distance preserves also the distance between x and y. Obviously, 1 ⊂eq Dn(F) ⊂eq An(F). We prove: An(C) ⊂eq d>0: d2 ∈ Q ⊂eq D2(F). Let K be a subfield of a commutative field Gamma extending C. Let psi2: Gamma2 × Gamma2->Gamma, psi2((x1,x2),(y1,y2))=(x1-y1)2+(x2-y2)2. We say that f:C2->K2 preserves unit distance if for each X,Y ∈ C2 psi2(X,Y)=1 implies psi2(f(X),f(Y))=1. We prove: if X,Y ∈ C2, psi2(X,Y) ∈ Q and X ≠ Y, then there exists a finite set S(X,Y) with X,Y ⊂eq S(X,Y) ⊂eq C2 such that any map f:S(X,Y)->K2 that preserves unit distance satisfies psi2(X,Y)=psi2(f(X),f(Y)) and f(X) ≠ f(Y).
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