Discrete versions of the Beckman-Quarles theorem

Abstract

TO APPEAR IN AEQUATIONES MATHEMATICAE - WITHOUT THEOREM 2. THEOREM 2 IS CORRECTLY PROVED IN PREVIOUS VERSIONS 1 AND 2. AUTHOR&#39;S VERSION 3 (WITH A NEW FIGURE 6A) IS UNNECESSARY. Let F \subseteq R denote the field of numbers which are constructible by means of ruler and compass. We prove that: (1) if x,y \in R^n (n>1) and |x-y| is an algebraic number then there exists a finite set S(x,y) \subseteq R^n containing x and y such that each map from S(x,y) to R^n preserving all unit distances preserves the distance between x and y; if x,y \in F^n then we can choose S(x,y) \subseteq F^n, (2) only algebraic distances |x-y| have the property from item (1), (3) if X1,X2,...,Xm \in R^n (n>1) lie on some affine hyperplane then there exists a finite set L(X1,X2,...,Xm) \subseteq R^n containing X1,X2,...,Xm such that each map from L(X1,X2,...,Xm) to R^n preserving all unit distances preserves the property that X1,X2,...,Xm lie on some affine hyperplane, (4) if J,K,L,M \in R^n (n>1) and |JK|=|LM| (|JK|<|LM|) then there exists a finite set C(J,K,L,M) \subseteq R^n containing J,K,L,M such that any map f:C(J,K,L,M) \to R^n that preserves unit distance satisfies |f(J)f(K)|=|f(L)f(M)| (|f(J)f(K)|<|f(L)f(M)|).