Rational distances from given rational points in the plane

Abstract

In this paper we study sets of points in the plane with rational distances from r prescribed points P_1, ...,P_r. A crucial case arises for r = 3, where we provide simple necessary and sufficient conditions for the density of this set in the real topology. We show in the Main Theorem that these conditions can be checked effectively (via congruences), proving that a related class of K3 surfaces satisfies the local-global principle. In particular, these conditions are always satisfied when P_1, P_2, P_3 are rational. This result completes and goes beyond the analysis of Berry, who worked under stronger assumptions, not always fulfilled for instance in all the cases where P_1, P_2, P_3 are rational. On the other hand, for r\ge 4, we show that points with rational distances correspond to rational points in a surface of general type, hence conjecturally not Zariski dense. However, at the present, we lack methods to prove this, given the fact that the surface is simply-connected, as we shall show. We give explicit proofs as well as describe in detail the geometry of the surfaces involved. In addition we discuss certain analogues for points with distances in certain ring of integers.