论文标题
有限点配置和分形设置中的常规值定理
Finite Point Configurations and the Regular Value Theorem in a Fractal setting
论文作者
论文摘要
在本文中，我们研究了两个问题，这些问题是由紧凑型集合$ e \ subset \ mathbb {r}^d $生成的有限点配置集的大小。第一个问题涉及Lebesgue测量或有限点配置集的Hausdorff尺寸取决于$ E $的尺寸。特别是，我们表明，如果平面套件的尺寸超过$ \ frac {5} {4} $，则存在E $中的点$ x \，以便对于每个整数$ k \ geq2 $，$ x $ x $的“ $ k $ chains”集合的“ $ k $ chains”套件具有正面的lebesgue措施。 第二个问题是对ERDS单位距离问题的连续类似物，该问题旨在确定可以在$ E $中显示带有规定间隙的点配置的最大次数。例如，给定一个带有规定侧面的三角形，并给出了一个足够规则的平面设置$ e $，带有hausdorff尺寸不少于$ \ frac {7} {4} $，我们表明，$ e $形成三角形的顶点的尺寸不超过$ 3 \ \ dim dim _ dim _ dim _ dim _ _ $ _ =除了欧几里得规范之外，我们还考虑了满足所谓的phong-stein旋转曲率条件的功能给出的更多一般距离。我们还探索了许多示例，以证明我们的结果在多大程度上。
In this article, we study two problems concerning the size of the set of finite point configurations generated by a compact set $E\subset \mathbb{R}^d$. The first problem concerns how the Lebesgue measure or the Hausdorff dimension of the finite point configuration set depends on that of $E$. In particular, we show that if a planar set has dimension exceeding $\frac{5}{4}$, then there exists a point $x\in E$ so that for each integer $k\geq2$, the set of "$k$-chains" with initial point at $x$ has positive Lebesgue measure. The second problem is a continuous analogue of the Erdős unit distance problem, which aims to determine the maximum number of times a point configuration with prescribed gaps can appear in $E$. For instance, given a triangle with prescribed sides and given a sufficiently regular planar set $E$ with Hausdorff dimension no less than $\frac{7}{4}$, we show that the dimension of the set of vertices in $E$ forming said triangle does not exceed $3\,\dim_{\mathcal{H}} (E)-3$. In addition to the Euclidean norm, we consider more general distances given by functions satisfying the so-called Phong-Stein rotational curvature condition. We also explore a number of examples to demonstrate the extent to which our results are sharp.
