EXPANSION OF THE BECKMAN-QUARLES THEOREM FOR RATIONAL SPACES ON MAPPINGS OF Q^d to Q^(d+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], (1953), 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, 4, 6, 8, 9, 10, 11 and 12], 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". Joseph Zaks, W.Benz, Karin B Chilakamarri, Robert Connelly and others, has found some of these values of d, until the year of 2005, me, Wafiq Hebi [5], in my ph.D thesis, proved that the rational analogues of the Beckman-Quarles theorem is true for all the dimensions  ≥5.

The purpose of this paper is to show that in some special dimension d, every unit distance-preserving mapping is an isometry.

In addition, which are the dimension  for which ?.