This is the second in my series of posts concerning lower bound for incidence problems (see the first post here). This post describes Elekes’ point-line construction, which is quite different from Erdős’ construction, although it yields asymptotically the same number of incidences. While preparing this post, I was quite surprised to discover that this construction yields constants better than those that are stated as the best known ones (e.g., in the research problems book).
Update 1 (11/26/2014): Frank the Zeeuw pointed out that this bound was already noticed in Section 1.1 of Roel Apfelbaum’s Ph.D. dissertation.
Update 2 (01/01/2018): It was pointed in this paper by Balko, Cibulka, and Valtr that the analysis of Pach and Tóth leads to the much better constant 1.27. This stronger bound is not stated in that paper due to a miscalculation.
Elekes’ construction is simpler and more elementary, in the sense that it only requires basic counting techniques (unlike Erdős’ construction which also relies on some number theory). As we shall see in the following posts, it is also easier to generalize to other types of planar curves. Moreover, unless I have some mistake, it seems to lead to better constants. Specifically, given points and lines in the plane, the maximum number of incidences, as shown by Pach and Tóth, is at least . In various places (such as the aforementioned open problems book) this is stated as the best known lower bound. The best known upper bound, obtained by combining a technique from the same paper of Pach and Tóth with a recent improvement by Ackerman, is about . Elekes’ construction leads to the improved lower bound .
In both constructions the point set is a subset of the integer lattice. However, Erdős’ construction is based on a square lattice (i.e., of size ), while Elekes’ contruction is a rather uneven lattice (except for the extreme case when ).
To simply the explanation, we set and . We define the point set to be
We define the line set to be
Notice that we indeed have
Consider a line that is defined by the equation , for some constants . Notice that for any value of in there exists a corresponding value of in the range , such that the point is incident to . That is, every line of is incident to exactly points of , and thus
And that is the end of the construction! In the next post of the series, I plan to start surveying planar configurations of other types of curves.