The Beckman–Quarles theorem for rational spaces Original Research Article

The Beckman–Quarles theorem states that every unit-preserving mapping from Rd to itself is an isometry, for all d⩾2. The analogues for the rational spaces Qd were established for all even dimensions, d,d⩾6, as well as for all odd dimensions d of the form d=2n2−1=m2, for integers n,m⩾2. The purpose of this paper is to present a proof of the rational analogues of the Beckman–Quarles Theorem in dimensions d of the form d=2n2−1, for all n⩾3. The proof is also applicable in all the even dimensions d of the form d=4k(k+1), for k⩾1, and in the real cases for all the dimensions d,d⩾3.