Breaking the 3/2 Barrier for Unit Distances in Three Dimensions
Joshua Zahl · International Mathematics Research Notices · 2018
Abstract We prove that every set of n points in $\mathbb{R}^{3}$ spans O(n295/197+ε) unit distances. This is an improvement over the previous bound of O(n3/2), which was a natural barrier for this problem. A key ingredient in the proof is a new result for cutting circles in $\mathbb{R}^{3}$ into pseudo-segments.