A discrete form of the Beckman-Quarles theorem for mappings from ${mathbb{R}}^{{text{2}}} {left( {{mathbb{C}}^{{text{2}}} } right)}$ to ${mathbb{F}}^{{text{2}}},$ where ${mathbb{F}}$ is a subfield of a commutative field extending ${mathbb{R}}({mathbb{C}})$

摘要

Let ${mathbb{F}}$ be a subfield of a commutative field extending ${mathbb{R}}.$ Let $varphi _{n} :{mathbb{F}}^{n} times {mathbb{F}}^{n} to {mathbb{F}},varphi _{n} {left( {{left( {x_{1} , ldots x_{n} } right)},{left( {y_{1} , ldots y_{n} } right)}} right)} = {left( {x_{1} - y_{1} } right)}^{2} + ldots + {left( {x_{n} - y_{n} } right)}^{2}$ . We say that $f:{mathbb{R}}^{n} to {mathbb{F}}^{n}$ preserves distance d ≥ 0 if |x − y| = d implies $varphi _{n} {left( {f(x),f(y)} right)} = d^{2}$ for each $x,y in {mathbb{R}}^{n} $ . Let $A_{n} ({mathbb{F}})$ denote the set of all positive numbers d such that any map $f:{mathbb{R}}^{n} to {mathbb{F}}^{n}$ that preserves unit distance preserves also distance d. Let $D_{n} ({mathbb{F}})$ 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 $f:S_{{xy}} to {mathbb{F}}^{n}$ that preserves unit distance satisfies |x − y|2 = φ n (f (y), f (y)). Obviously, ${ 1} subseteq D_{n} ({mathbb{F}}) subseteq to A_{n} ({mathbb{F}})$ . We prove: $A_{n} ({mathbb{C}}) subseteq { d > 0:d^{2} in {mathbb{Q}}} subseteq D_{2} ({mathbb{F}})$ . Let ${mathbb{K}}$ be a subfield of a commutative field Γ extending ${mathbb{C}}$ . Let $psi _{2} :Gamma ^{2} times Gamma ^{2} to Gamma ,,,,psi _{2} {left( {(x_{1} ,x_{2} ),(y_{1} ,y_{2} )} right)} = (x_{1} - y_{1} )^{2} + (x_{2} - y_{2} )^{2}$ . We say that $f:{mathbb{C}}^{2} to {mathbb{K}}^{2}$ preserves unit distance if ψ2(X,Y) = 1 implies ψ2(f (X), f (Y)) = 1 for each $X,Y in {mathbb{C}}^{2}$ . We prove: if $X,Y in {mathbb{C}}^{2} ,psi _{2} (X,Y) in {mathbb{Q}}$ and X ≠ Y, then there exists a finite set S XY with ${ X,Y} subseteq S_{{XY}} subseteq {mathbb{C}}^{2}$ such that any map $f:S_{{XY}} to {mathbb{K}}^{2}$ that preserves unit distance satisfies ψ2(X, Y) = ψ2(f (X), f (Y)) and f (X) ≠ f (Y).