Reducing the distinct distances problem in $\mathbb{R}^d$ to an incidence problem

Place: Courant Institute, WWH1314

The distinct distances problem, posed by Erdős in 1946, asks: what is the minimum number of distinct distances a set of $n$ points in $\mathbb{R}^d$ can span? Guth and Katz showed in their 2015 paper that every set of $n$ points in $\mathbb{R}^2$ spans $\Omega(n/\log n)$ distances, which is asymptotically tight (up to a factor of $\sqrt{\log n}$). Their proof involved reducing the problem of counting distinct distances to a point-line incidence problem in $\mathbb{R}^3$. I will discuss work with Adam Sheffer extending this reduction to the distinct distances problem in $\mathbb{R}^d$.