Finite point configurations and the regular value theorem in a fractal setting

Abstract

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" 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 $3\,{\rm dim}_H (E)-3$. 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.