Distinct distances on the plane

Distinct distances on the plane

Frontiers In Mathematics Distinguished Lecture Series (First Lecture)

Hong Wang (IAS)

Friday, September 18, 2020 -
12:00pm to 1:00pm
Physics 119

Given \(N\) distinct points on the plane, what is the minimal number of distinct distances between them? This problem was posed by Paul Erdos in 1946 and essentially solved by Guth and Katz in 2010. We are going to consider a continuous analogue of this problem, the Falconer distance problem. Given a set \(E\) of dimension \(s>1\), what can we say about its distance set \(\Delta(E)=\{|x-y|: x, y\in E\}\)? Falconer conjectured in 1985 that \(\Delta(E)\) should have positive Lebesgue measure. In the recent years, people have attacked this problem in different ways (including geometric measure theory, Fourier analysis, and combinatorics) and made some progress for various examples and for some range of \(s\).

Last updated: 2020/09/24 - 6:14am