The Polynomial Method

The way we were thinking about this was to draw circles around the points under consideration, and noting that we expect a great many points to lie on the boundaries of other circles. But clearly, in that setting, we are not using all the information we have; in particular, we are never accounting for the fact that some of the circles may have equal radii, which would also decrease the value of t. How can we somehow encode this information? Suppose we denote our set of points in the plane by P ⊂ R; let d(P ) be the set of nonzero distances between points in P , and Q(P ) be a measure of how two different pairs of points can have the same nonzero distance between them. Formally speaking,