Erd\H{o}s Distance Problem in $\mathbb{R}^d$

Abstract

In this paper, we prove Erd\H{o}s distance conjecture in $\mathbb{R}^d$, namely, a set of $n$ points in $\mathbb{R}^2$ determines $\Omega(\frac{n}{\sqrt{\log n}})$ distances, and for $d\ge 3$, a set of $n$ points in $\mathbb{R}^d$ determines $\Omega(n^{\frac{2}{d}})$ distinct distances.