The Beckman-Quarles Theorem For Rational Spaces

Abstract

Let Rd and Qd 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 Rd or Qd and is called - distance preserving implies , for all x,y in . Let G(Qd,a) denote the graph that has Qd as its set of vertices, and where two vertices x and y are connected by edge if and only if . Thus, G(Qd,1) is the unit distance graph. Let ω(G) denote the clique number of the graph G and let ω(d) denote ω(G(Qd, 1)). The Beckman-Quarles theorem [1] states that every unit- distance-preserving mapping from Rd into Rd 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 Qd into Qd 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)→.