The Beckman-Quarles Theorem and some related topic
Joseph Zaks (Università di Haifa)
The Beckman-Quarles Theorem asserts that every unit preserving mapping f: R^d --> R^d is an isometry, provided d >= 2. There exists a unit preserving mapping f: R2 --> R6 which is not an isometry, and this can be extended for unit preserving mappings g: R^d --> R^s, for all s >= X(G(R,d,1)).
A lot of attention was given to unit preserving mappings f: Q^d --> Q^d for d >= 5, since it was known that such mappings for 2 <= d <= 4 need not be isometries. It is known now that they are isometries for all d >= 5.
A whole area of research is widely open, concerning r-distance preserving mappings f: Q^d --> Q^d, provided r2 is a rational number, and d is sufficiently large.
The last topic deals with the following result: If A is a subset in E^n for which all the mutual distances of points in A have their square being a rational number, then A is congruent to a subset of Q^s, where s is at most four (4) times the affine dimension of A.
In other words, a necessary and sufficient condition for a set in E^n to be congruent to a subset of Q^s, for some s, is that the square of every distance between points of A is a rational number.