The number of different distances determined by a set of points in the Euclidean plane

Abstract

In 1946 P. Erdos posed the problem of determining the minimum numberd(n) of different distances determined by a set ofn points in the Euclidean plane. Erdos provedd(n) ?cn1/2 and conjectured thatd(n)?cn/ ?logn. If true, this inequality is best possible as is shown by the lattice points in the plane. We showd(n)?n4/5/(logn)c.