An upper bound for the Erd\H{o}s unit distance problem in the plane

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Abstract

In this paper, using the method of compression, we prove a stronger upper bound for the Erd\H{o}s unit distance problem in the plane by showing that\begin{align}\# \bigg\{||\vec{x_j}-\vec{x_t}||:\vec{x}_t, \vec{x_j}\in \mathbb{E}\subset \mathbb{R}^2,~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j \leq n\bigg\}\ll_2 n^{1+o(1)}.\nonumber\end{align}

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europepmc
last seen: 2026-05-19T01:45:01.086888+00:00