A discrete form of the Beckman-Quarles Theorem for rational spaces

摘要

The Beckman-Quarles Theorem states that every unit-preserving mapping of R d into R d is an isometry. A. Tyszka proved that every unit preserving mapping of Q 8 into Q 8 is an isometry. We extend the last result by showing that every unit-preserving mapping from Q d into Q d is an isometry for all even d of the form $ d=4k(k+1), kgeq 1 $ , as well as for all odd d of the form $ 2k^{2}-1 $ which are complete squares, $ k geq 2 $ .