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arXiv · 1706.05118

Breaking the 3/2 barrier for unit distances in three dimensions

Joshua Zahl
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Abstract

We prove that every set of $n$ points in $\mathbb{R}^3$ spans $O(n^{295/197+ε})$ unit distances. This is an improvement over the previous bound of $O(n^{3/2})$. A key ingredient in the proof is a new result for cutting circles in $\mathbb{R}^3$ into pseudo-segments.

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Joshua Zahl. 2022-02-28. Breaking the 3/2 barrier for unit distances in three dimensions. https://doi.org/10.1093/imrn%2Frnx336

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