Jonathan Passant

Configurations and Erdős style distance problems

I will discuss generalisations of the Erdős distance problem to larger configurations. This work was begun by Solymosi-Tardos and Rudnev who proved sharp bound for the number of similar/congruent triangles. We use the work of Guth-Katz along with a new incidence bound to prove sharp (up to logs) results for the number of distance-distinct configurations of large numbers of points.

The start of the talk will give an introduction to distance problems in finite sets over the reals, including a quick set up of the Elekes-Sharir-Guth-Katz framework.

I will conclude with a discussion of the three-star problem, the case that stops one establishing a result over all configurations.