Today I wish to mention another new result related to distinct distances. This time it is this paper by Gasarch, Harris, Ulrich, and Zbarsky. I (or rather, google scholar) found this paper on Gasarch’s website (next to a disclaimer stating that this is a work in progress and might be incorrect). [Update (Aug 2013): as commented by David Conlon, these results are not in their final form, and an updated version including also David Conlon and Jacob Fox should appear in the not-too-distant future.] [Update #2 (Jan 2014): The updated version, containing six authors, can be found here.]
The paper contains various results, all related to a generalization of a family of problems that I recently surveyed. Following the notation in the paper, concerns distinct volumes of sets of -dimensional simplices (i.e., simplices that are determined by points) in . Specifically, denotes the maximum number satisfying that for every set of points in , with no points on a common -flat, there exists a subset of points that do not determine two -dimensional simplices with the same volume.
For example, the case of is about distinct distances in the plane. That is, is the maximum number such that any set of points in contains a subset of points that do not determine any distance more than once. Thus, the proof of Charalambides (which I also described here) implies the bound .
As another example, is the maximum number such that any set of points in , with no three collinear, contains a subset of points that do not determine two triangles with the same area. Consider the set of vertices of a regular -gon, as in the following figure.
Such a set does not contain three collinear points, and any triangle area appears times. This immediately implies (since any asymptotically larger subset with no repeated triangle areas would determine distinct areas). A somewhat similar result by Iosevich, Roche-Newton, and Rudnev states that for any set of points in the plane (no three collinear), and for any point , there are distinct areas of triangles that are spanned by together with two other points of .
So what are the new results in the current paper? First, the authors generalize Charalambides’ bound for to higher dimensions. Specifically, by induction on the dimension (with Charalambides’ bound for as the induction base), they obtain
This slightly improves Thiele’s bound of . Moreover, the current paper seems to be the first to study the case of for . Specifically, it derives the bound
As in other recent related papers, the authors rely on tools from algebraic geometry to obtain this result. The paper also studies cases where is infinite, but I will not discuss these in this post.