On Distinct Distances and Incidences: Elekes's Transformation and the New Algebraic Developments ∗
Micha Sharir · 2010
We first present a transformation that Gyuri Elekes has devised, about a decade ago, from the celebrated problem of Erdős of lower-bounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. Elekes has offered conjectures involving the new setup, which, if correct, would imply that the number of distinct distances in an s-element point set in the plane is always Ω(s/log s). Unfortunately, these conjectures are still not fully resolved. We then review the recent progress made on the transformed incidence problem, based on a new algebraic approach, originally introduced by Guth and Katz. Full details of the results reviewed