Distances in Planar Integral Point Sets

Abstract

We show that very small distances in a planar integral point set are essentially one-dimensional. Let P be a non-collinear set of n points in the plane, all of whose pairwise distances are integers. We prove that, for a sufficiently small $c>0$, at most one pair of points can determine a distance below $n^{c\log\log{n}}$ unless all such short pairs are supported on a single line. We also give a construction showing that the line-supported alternative is necessary: there are arbitrarily large non-collinear integral point sets with one off-line point and with the first two distinct distances 2 and 4. We conjecture that in the general case (no three points are collinear) even the smallest distance should be large, at least $n^{c\log\log{n}}$, however we can prove a linear lower bound only.