The higher-dimensional Erdos distinct-distances conjecture

Theorem 1.1: for every integer $d\ge3$ there is $c_d>0$ such that every set of $n\ge2$ distinct points in $\mathbb R^d$ determines at least $c_dn^{2/d}$ distinct distances, matching the integer grid up to the constant. No dependence of $c_d$ on $d$ is made explicit, nothing is claimed for pinned distances (many distances from one point), and the planar problem, whose conjectured order is $n/\sqrt{\log n}$, is untouched beyond reusing the Guth-Katz bound. Priority: Aksoy Yazici earlier claimed the same constant-factor bound in all dimensions $d\ge3$, but withdrew the manuscript because a step was wrong; it is not an opposite claim.

Theorem 1.1: for every integer $d\ge3$ there is $c_d>0$ such that every set of $n\ge2$ distinct points in $\mathbb R^d$ determines at least $c_dn^{2/d}$ distinct distances, matching the integer grid up to the constant. No dependence of $c_d$ on $d$ is made explicit, nothing is claimed for pinned distances (many distances from one point), and the planar problem, whose conjectured order is $n/\sqrt{\log n}$, is untouched beyond reusing the Guth-Katz bound. Priority: Aksoy Yazici earlier claimed the same constant-factor bound in all dimensions $d\ge3$, but withdrew the manuscript because a step was wrong; it is not an opposite claim.