论文标题
在非式表面和圆之间的不同距离
Distinct distances on non-ruled surfaces and between circles
论文作者
论文摘要
我们在$ {\ Mathbb r}^3 $上的非式代数表面上的不同距离上提高了最佳界限。特别是，我们表明，对于任何$ \ varepsilon> 0 $，$ n $点上的$ n $点（n^{n^{32/39- \ varepsilon} \ right）$不同的距离。我们的证明适应了基于穿越引理的平面案例的Székelly证明。 作为在表面上不同距离的证据的一部分，我们还为$ {\ Mathbb r}^3 $之间的圆圈之间的不同距离获得了新的结果。考虑点集$ {\ MATHCAL P} _1 $和$ {\ MATHCAL P} _2 _2 $的尺寸$ M $和$ n $，因此每个集合都位于$ {\ Mathbb r}^3 $中的独特圆上。我们表征了两组之间不同距离的数量可能为$ O（m+n）$时的情况。这包括具有少量距离的新配置。在任何其他情况下，我们都证明不同距离的数量为$ω\ left（\ min \ left \ {m^{2/3} n^{2/3}，m^2，n^2 \ right \} \ right）$。
We improve the current best bound for distinct distances on non-ruled algebraic surfaces in ${\mathbb R}^3$. In particular, we show that $n$ points on such a surface span $Ω\left(n^{32/39-\varepsilon}\right)$ distinct distances, for any $\varepsilon>0$. Our proof adapts the proof of Székely for the planar case, which is based on the crossing lemma. As part of our proof for distinct distances on surfaces, we also obtain new results for distinct distances between circles in ${\mathbb R}^3$. Consider point sets ${\mathcal P}_1$ and ${\mathcal P}_2$ of respective sizes $m$ and $n$, such that each set lies on a distinct circle in ${\mathbb R}^3$. We characterize the cases when the number of distinct distances between the two sets can be $O(m+n)$. This includes a new configuration with a small number of distances. In any other case, we prove that the number of distinct distances is $Ω\left(\min\left\{m^{2/3}n^{2/3},m^2,n^2\right\}\right)$.
