The rational analogue of the Beckman-Quarles Theorem and the rational realization of some sets in E

We describe the recent developments concerning the rational analogues of the Beckman-Quarles Theorem, and discuss a related result concerning isometric embeddings in Q of subsets of E. 1 – Let E denote the Euclidean d-space, and let Q denote the Euclidean rational d-space. A mapping f : E → E is called ρ-distance preserving if ‖x − y‖ = ρ implies that ‖f(x)−f(y)‖ = ρ. The Beckman Quarles Theorem [1] asserts that every mapping f : E → E which preserves unit distance is an isometry, provided d ≥ 2; for a discrete version, see Tyszka [9]. W. Benz [2, 3] and H. Lenz [7] noticed that if d = 2, 3 or 4, a unit-distance preserving mapping from Q into Q needs not be an isometry. A Tyszka [10] showed that every unit distance preserving mapping f : Q → Q is an isometry. In a sequence of papers [12,13] we extended these results to all even dimensions d of the form d = 4k(k+1) and all the odd dimensions d of the form d = 2m − 1. W. Benz [2, 3] had shown that every mapping f : Q → Q which preserves the distances 1 and 2 (or, equivalently, 1 and n, n ≥ 2) is an isometry, provided d ≥ 5. We [14] had shown that every mapping f : Q → Q which preserves the distances 1 and √ 2 is an isometry, provided d ≥ 5. R. Connelly and J. Zaks [5]