论文标题
统一的分布和几何发生率理论
Uniform distribution and geometric incidence theory
论文作者
论文摘要
由于ERD \ H OS引起的著名单位距离猜想说，在欧几里得飞机中的$ n $点之间，单位距离不能超过$C_εn^{1+ε} $ times（对于任何$ε> 0 $）（例如，请参见f。\ cite \ cite {sst84}和其中包含的参考文献）。在三个维度中，猜想的界限为$ cn^{\ frac {4} {3}}} $（参见例如\ cite {kmss12}和\ cite {z19}）。在四个及更高的尺寸中，此问题在其一般公式中失去了含义，因为镜头示例表明，可以在尺寸$ 4 $中构建一组$ n $点的点，而在单位距离出现$ \ of \ bout n n^2 $ times的位置（例如，见例如\ cite {b97}））。但是，镜头的示例本质上是一维，这增加了单位距离猜想在点集的其他结构假设下仍然非常有趣的可能性。在\ cite {i19}，\ cite {is16}，\ cite {imt12}，\ cite {iru14}，\ cite {oo15}中探索了这一观点，并导致了单位距离问题与持续的连接之间的一些有趣的连接，尤其是falconer的距离，cite {oo15} coite {oo15} falconer距离（falconer）。 在本文中，我们研究了单位距离问题及其变体，假设基本的点集均匀分布。我们在这种情况下证明了几个发病率界限，并在组合几何形状的发病率问题中阐明了均匀分布的序列的一些关键特性。
A celebrated unit distance conjecture due to Erd\H os says that that the unit distances cannot arise more than $C_εn^{1+ε}$ times (for any $ε>0$) among $n$ points in the Euclidean plane (see e.g. \cite{SST84} and the references contained therein). In three dimensions, the conjectured bound is $Cn^{\frac{4}{3}}$ (see e.g. \cite{KMSS12} and \cite{Z19}). In dimensions four and higher, this problem, in its general formulation, loses meaning because the Lens example shows that one can construct a set of $n$ points in dimension $4$ and higher where the unit distance arises $\approx n^2$ times (see e.g. \cite{B97}). However, the Lens example is one-dimension in nature, which raises the possibility that the unit distance conjecture is still quite interesting in higher dimensions under additional structural assumptions on the point set. This point of view was explored in \cite{I19}, \cite{IS16}, \cite{IMT12}, \cite{IRU14}, \cite{OO15} and has led to some interesting connections between the unit distance problem and its continuous counterparts, especially the Falconer distance conjecture (\cite{Falc85}). In this paper, we study the unit distance problem and its variants under the assumption that the underlying family of point sets is uniformly distributed. We prove several incidence bounds in this setting and clarify some key properties of uniformly distributed sequences in the context of incidence problems in combinatorial geometry.
