欧几里得空间薄子集中单形的一种初等方法
An elementary approach to simplexes in thin subsets of Euclidean space
摘要：
证明了如果$E \subset {\Bbb R}^d$的Hausdorff维数$d \ge 3$大于$\min \left\{ \frac{dk+1}{k+1}, \frac{d+k}{2} \right\},$，则$k$ -维单形与$E$中顶点的同余类集合$T_k(E)$的${k+1 \choose 2}$ -维Lebesgue测度是正的.这改进了先前已知的最佳界限，通过不同且概念上更简单的方法将Erdoğan-Hart-Iosevich（2012）中获得的$\frac{d+k+1}{2}$阈值降低到$\frac{d+k}{2}$。我们还给出了Greenleaf-Iosevich（2012），Grafakos-Greenleaf-Iosevich-Palsson（2015）中获得的$d$维单形的$d-\frac{d-1}{2d}$阈值的一个更简单的证明。
英文摘要：
We prove that if the Hausdorff dimension of $E \subset {\Bbb R}^d$, $d \ge 3$, is greater than $\min \left\{ \frac{dk+1}{k+1}, \frac{d+k}{2} \right\},$ then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$, the set of congruence classes of $k$-dimensional simplexes with vertices in $E$, is positive. This improves the best bounds previously known, decreasing the $\frac{d+k+1}{2}$ threshold obtained in Erdoğan-Hart-Iosevich (2012) to $\frac{d+k}{2}$ via a different and conceptually simpler method. We also give a simpler proof of the $d-\frac{d-1}{2d}$ threshold for $d$-dimensional simplexes obtained in Greenleaf-Iosevich (2012), Grafakos-Greenleaf-Iosevich-Palsson (2015).
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