论文标题
在ERDS距离问题上
On the Erdős distance problem
论文作者
论文摘要
在本文中，使用压缩方法，我们为ERDS单位距离问题恢复了下限，并为不同的距离猜想提供了替代证明。特别是，在$ \ mathbb {r}^k $中，所有$ k \ geq 2 $，我们都有\ begin {align} \＃\ bigG \ {|| \ vec {|| \ vec {x_j} - \ \ vec {x_t} || t，j \ leq n，〜\ vec {x_j}，〜\ \ vec {x} _t \ in \ mathbb {r}^k \ bigG \} \ geq c \ geq c \ frac {\ sqrt {\ sqrt {k}}} $ C> 0 $。我们还表明\ begin {align} \＃\ big {d_j：d_j = || \ vec {x_s} - \ \ vec {y_t} ||，〜d_j \ neq d_i，〜1 d \ frac {\ sqrt {k}} {2} n^{\ frac {2} {k} {k} -o（1）} \ nonumber \ end dend {align} in Some $ d> 0 $。这些下限概括了ERDS单位距离的下限，并将不同的距离问题与更高的尺寸相关。
In this paper, using the method of compression, 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 $k\geq 2$, we have \begin{align} \# \bigg\{||\vec{x_j}-\vec{x_t}||:~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j \leq n,~\vec{x_j},~\vec{x}_t \in \mathbb{R}^k\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 generalizes the lower bounds of the Erdős unit distance and the distinct distance problem to higher dimensions.
