On the Erdős distance problem

Abstract

In this paper, using the compression method, we recover the lower bound for the Erdős unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in ${\mathbb{R}}^k$ for all $k\ge 2$, we have \begin{align} \#\bigg\{(\vec{x}_t,\vec{x_j})\in \mathbb{E}\subset\mathbb{R}^k~:~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j\leq n\bigg\}\geq C\frac{\sqrt{k}}{2}n^{1+o(1)}\nonumber \end{align} for some $C>0$. We also show that \begin{align} \# \bigg\{d_j:d_j=||\vec{x_s}-\vec{y_t}||,~d_j\neq d_i,~1\leq s,t\leq n\bigg\}\geq D\frac{\sqrt{k}}{2}n^{\frac{2}{k}-o(1)}\nonumber \end{align} for some $D>0$. These lower bounds generalize the lower bounds of the Erdős unit distance and the distinct distance problem to higher dimensions.

11 pages; the paper has been reformatted and the introduction greatly expanded; the ideas remain unchanged; arXiv admin note: text overlap with arXiv:2106.15621, arXiv:1912.08075