, Volume 59, Issue 1-2, pp 124-133

Discrete versions of the Beckman—Quarles theorem

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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 \( |f(J)f(K)|=|f(L)f(M)| (|f(J)f(K)| < |f(L)f(M)|) \) .

Received: May 5, 1998; revised version: April 10, 1999.