Discrete versions of the Beckman—Quarles theorem

Summary.

We prove that:¶(1) if $x,y\in {\Bbb R}^{n}\ (n>1)$ and $|x-y|$ is an algebraic number then there exists a finite set $S_{xy}\subseteq {\Bbb R}^{n}$ containing x and y such that each map from S _xy to ${\Bbb R}^{n}$ preserving all unit distances preserves the distance between x and y,¶(2) only algebraic distances $|x-y|$ have the property from item (1),¶(3) if $X_{1},X_{2},\dots ,X_{m}\in {\Bbb R}^{n}\ (n>1)$ lie on some affine hyperplane then there exists a finite set $L(X_{1},X_{2},\dots ,X_{m}) \subseteq {\Bbb R}^{n}$ containing $X_{1},X_{2},\dots ,X_{m}$ such that each map from $L(X_{1},X_{2},\dots ,X_{m})$ to ${\Bbb R}^{n}$ preserving all unit distances preserves the property that $X_{1},X_{2},\dots ,X_{m}$ lie on some affine hyperplane,¶(4) if $J,K,L,M \in {\Bbb R}^{n}\ (n>1)$ and $|JK|=|LM|\,(|JK|T < |LM|)$ then there exists a finite set $C_{JKLM}\subseteq {\Bbb R}^{n}$> containing J,K,L,M such that any map $f : C_{JKLM} \rightarrow {\Bbb R}^{n}$ that preserves unit distances satisfies $|f(J)f(K)|=|f(L)f(M)| (|f(J)f(K)| < |f(L)f(M)|)$.