A discrete form of the Beckman-Quarles theorem for rational eight-space

Let Q be the field of rationals numbers. We prove that: (1) if x, y ∈ R (n > 1) and |x− y| is constructible by means of ruler and compass then there exists a finite set Sxy ⊆ R containing x and y such that each map from Sxy to R n preserving unit distance preserves the distance between x and y, (2) if x, y ∈ Q then there exists a finite set Sxy ⊆ Q containing x and y such that each map from Sxy to R 8 preserving unit distance preserves the distance between x and y. Theorem 1 may be viewed as a discrete form of the classical Beckman-Quarles theorem, which states that any map from R to R (2 ≤ n < ∞) preserving unit distance is an isometry, see [1]-[3]. Theorem 1 was announced in [9] and prove there in the case where n = 2. A stronger version of Theorem 1 can be found in [10], but we need the elementary proof of Theorem 1 as an introduction to Theorem 2. Theorem 1. If x, y ∈ R (n > 1) and |x− y| is constructible by means of ruler and compass then there exists a finite set Sxy ⊆ R containing x and y such that each map from Sxy to R n preserving unit distance preserves the distance between x and y. Proof. Let us denote by Dn the set of all non-negative numbers d with the following property: if x, y ∈ R and |x − y| = d then there exists a finite set Sxy ⊆ R such that x, y ∈ Sxy and any map f : Sxy → R that preserves unit distance preserves also the distance between x and y. Obviously 0, 1 ∈ Dn. We first prove that if d ∈ Dn then √ 2 + 2/n · d ∈ Dn. Assume that d > 0, x, y ∈ R and |x − y| = √ 2 + 2/n · d. Using the notation of Figure 1 we show that Sxy := {Sab : a, b ∈ {x, y, ỹ, p1, p2, ..., pn, p̃1, p̃2, ..., p̃n}, |a − b| = d} satisfies the condition of Theorem 1. Figure 1 shows the case n = 2, but equations below Figure 1 describe the general case n ≥ 2; z denotes the centre of the (n− 1)dimensional regular simplex p1p2...pn. AMS (1991) Subject Classification: Primary 51M05, Secondary 05C12