Falconer型关联定理和点构形定理的维数下界
Dimensional lower bounds for Falconer type incidence and point configuration theorems
摘要：
设$1 \leq k \leq d$并考虑子集$E\subset \mathbb{R}^d$。本文研究了$E$的Hausdorff维数必须有多大，才能使$E$中不同的非全等$k$ -单形（即来自$E$的$k+1$点的非全等点构型）的集合具有正的Lebesgue测度.这推广了$k=1$的情况下，著名的Falconer距离问题和一个主要的开放问题的几何测量理论。我们建立了$k$ -单形的Falconer型定理的一个维数下限$\frac{d(k+1)}{d+2}$.该阈值在$d/2 \leq k \leq d$范围内是非平凡的，并且通过使用高斯圆问题的结果在标准格中计数单形来获得。许多关于Falconer型定理的结果都是通过关联定理建立的，这些关联定理一般都为点构形定理建立了充分但非必要的条件。通过推广Mattila的一个例子，我们还建立了$k$ -单形的关联定理的$\frac{d+1}{2}$的维数下限，其中$k\leq d \leq 2k+1$.最后，我们证明了在任何大于$3$的维数下，凸集上三角形的关联定理的维数下限为$\frac{d+1}{2}$。这最后一个结果概括了Iosevich和Senger在Valtr的建筑上建立的距离上的工作。最后的结果利用数论机器估计的解决方案的丢番图方程的数量。
英文摘要：
Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is, noncongruent point configurations of $k+1$ points from $E$) to have positive Lebesgue measure. This generalizes the $k=1$ case, the well-known Falconer distance problem and a major open problem in geometric measure theory. We establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer type theorems for $k$-simplices. This threshold is nontrivial in the range $d/2 \leq k \leq d$ and is obtained through counting simplices in a standard lattice using results of the Gauss circle problem. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We also establish a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for $k$-simplices where $k\leq d \leq 2k+1$ by generalizing an example of Mattila. Finally, we prove a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for triangles in a convex setting in every dimension greater than $3$. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.
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