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 (For this discussion it will not be important whether we count the pair separately from .) 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 equally spaced collinear points with unit spacing, one can obtain a linear construction with . Erdös observed that one can improve this construction asymptotically: Theorem 1 (Erdös construction) There exists point sets of arbitrarily large cardinality such that for some absolute constant . In fact, in the construction one could take arbitrarily close to . Erdös famously asked whether had to be bounded above by ; and for decades there was significant effort expended on upper bounding , with the best known…