Hong Wang, Institute for Advanced Study

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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 analog of this problem: the Falconer distance set problem. Given a set $E$ of Hausdorff dimension $s>d/2$ in $\mathbb{R}^d$ , Falconer conjectured that its distance set $\Delta(E)=\{ |x-y|: x, y \in E\}$ should have positive Lebesgue measure. In recent years, people have studied this problem using different techniques in geometric measure theory and Fourier analysis. We are going to study a couple of examples and discuss how people solve the examples using those techniques.