Erdős Distance Problem in ${\mathbb{R}}^d$

Abstract

In this paper, we prove Erdős 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.

The paper is withdrawn as the proof of Theorem 1.2 is not correct