A discrete form of the Beckman-Quarles theorem for rational spaces (Q1601387)

Let \(A=\{4k(k+1):k=1,2,3,\dots \} \cup (\{k^2: k=1,2,3,\dots \} \cap \{2k^2-1: k=2,3,4,\dots \})\). The author proves: if \(n \in A\) and \(x,y \in \mathbb Q^n\), then there exists a finite set \(\{x,y\} \subseteq S_{xy} \subseteq \mathbb Q^n\) such that for every unit-distance preserving mapping \(f:S_{xy} \to \mathbb Q^n\), \(\|f(x)-f(y)\|=\|x-y\|\) holds. The theorem for \(n=8\) was earlier proved by \textit{A. Tyszka} [Aequationes Math. 62, No. 1-2, 85-93 (2001; Zbl 1004.51023)].