During 2013, significant progress has been obtained for several problems that are related to the Erdos distinct distances problem. In this talk I plan to briefly describe some of these results and the tools that they rely on. I will focus on the following two results.

Let P and P´ be two sets of points in the plane, so that P is contained in a line L, P´ is contained in a line L´, and L and L´ are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs of PxP´ is \Omega(\min{|P|^{2/3}|P´|^{2/3},|P|^2, |P´|^2}). In particular, if |P|=|P´|=m, then the number of these distinct distances is \Omega(m^{4/3}), improving upon the previous bound \Omega(m^{5/4}) of Elekes.

In the seoncd result, we study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distances, then no line contains \Omega(n^{7/8}) points of P and no circle contains \Omega(n^{5/6}) points of P.

In both cases, we rely on a bipartite and partial variant of the Elekes-Sharir framework, which has been used by Guth and Katz in their 2010 solution of the general distinct distances problem. We combine this framework with some basic algebraic geometry, with a theorem from additive combinatorics by Elekes, Nathanson, and Ruzsa, and with a recent incidence bound for plane algebraic curves by Wang, Yang, and Zhang.

The first result is a joint work Micha Sharir (Tel Aviv) and József Solymosi (UBC). The second is a joint work with Joshua Zahl (MIT) and Frank de Zeeuw (EPFL).