Jacob Fox, Janos Pach, and Andrew Suk just placed this paper on arXiv. The paper seems very nice, but unfortunately for Ben Lund, Frank de Zeeuw, and myself, it contains a result that is almost identical to one that we had but did not yet publish. Instead of sulking alone in the dark, in this post I describe the problem and our result.
Let denote the minimum number of distinct distances that can be determined by a set of points with the property that any points of determine at least distinct distances. That is, by assuming that the point set satisfies a local property, we wish to conclude the global property of many distinct distances.
For example, the value of is the minimum number of distinct distances that are determined by a set of points that do not span any isosceles triangles (including degenerate triangles with three collinear vertices). Since no isosceles triangles are allowed, every point determines distinct distances with the other points of the set, and we thus have . Erdős observed the following upper bound for . Behrend proved that there exists a set of positive integers , such that no three elements of determine an arithmetic progression and that . Therefore, the point set does not span any isosceles triangles. Since and , we have .
In the same paper, Erdős derived the more general bound
As Fox, Pach, and Suk point out, a result by Sarkozy and Selkow implies the slightly stronger (with depending on )
We are now ready to state our result. The result of Fox, Pach, and Suk is identical up to the being replaced with . While there are some similarities between the two proofs, they do not seem to be identical.
Theorem 1. For every even and , we have
Proof. We begin the proof exactly as in the Elekes-Sharir framework. Consider a set of points and let
Let denote the number of distinct distances that are determined by , and denote these distances as . Let . Since every pair of is in exactly one , we have . By the Cauchy-Schwarz inequality, we get
For a pair of points we define a point , and a hypersurface that is the zero set of
Let and , so . Notice that there is a bijection between the incidences in and the quadruples . Thus, to derive an upper bound for it suffices to obtain an upper bound on .
Assume that there is a copy of in the incidence graph of . This means that there is a set of points such that for every and . Thus, if every points of determine at least distances, the incidence graph contains no copy of . The following is Theorem 1.2 from a paper by Fox, Pach, Suk, Zahl, and myself.
Theorem 2. Let be a set of points and let be a set of constant-degree algebraic varieties, both in , such that the incidence graph of does not contain a copy of (for constants ). Then for every , we have
By applying Theorem 2 with , , , and , we obtain
Combining this with yields the assertion of the theorem.