A digestion of unit distance constructions

Suppose that one has a set of points in the plane, which we will think of as the complex plane . Let denote the number of unit distances determined by these points, i.e., pairs of points whose displacement obeys the equation (It makes little difference for the asymptotics, but we will count the pair separately from here.) The Erdös unit distance problem asks, for a given large number , what is the largest possible value of amongst all sets of cardinality ? For instance, if one takes to be equall...