The Beckman-Quarles Theorem For Rational Spaces

Abstract

Let R^d and Q^d denote the real and the rational d-dimensional space, respectively, equipped with the usual Euclidean metric. For a real number, a mapping  where X is either R^d or Q^d and is called - distance preserving  implies  , for all x,y in .

Let G(Q^d,a) denote the graph that has Q^d  as its set of vertices, and where two vertices x and y are connected by edge if and only if  . Thus, G(Q^d,1) is the unit distance graph. Let ω(G) denote the clique number of the graph G and let ω(d) denote ω(G(Q^d, 1)).

The Beckman-Quarles theorem [1] states that every unit- distance-preserving mapping from R^d into R^d is an isometry, provided d ≥ 2.

The rational analogues of Beckman- Quarles theorem means that, for certain dimensions d, every unit- distance preserving mapping from Q^d into Q^d is an isometry.

A few papers [2, 3, 4, 5, 6, 8,9,10 and 11] were written about rational analogues of this theorem, i.e, treating, for some values of  the property "Every unit- distance preserving mapping  is an isometry".

The purpose of this section is to prove the following Lemma

Lemma: If x and y are two points in  so that:

where  then there exists a finite set S(x,y), contains x and y such that f(x)≠f(y) holds for every unit- distance preserving mapping  f: S(x,y)→.