The Beckman-Quarles Theorem For Rational Spaces: Mapping Of Q^d To Q^d That Preserve Distance 1

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 thesis is to present all the results (see [3, 5, 6 and 7]) about the rational analogues of the Beckman-Quarles theorem, and to establish rational analogues of the Beckman-Quarles theorem, for all the dimensions  ≥5.