Seminar Event Detail


Combinatorics

Date:  Friday, November 22, 2013
Location:  3866 East Hall (4:10 PM to 5:00 PM)

Title:  Recent progress in distinct distances problems

Abstract:   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 second 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).

Files:


Speaker:  Adam Sheffer
Institution:  Tel Aviv

Event Organizer:     

 

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