Geometry

An Upper Bound for the Erd\h{o}s Unit Distance Problem in the Plane

Authors: Theophilus Agama
Comments: 8 Pages. A few technicalities resolved regarding the scale of compression and the inequality in the notion of points contained in a compression ball has been made strict. This is because the case of equality is treated separately as admissible points.

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}
Category: Geometry