2020/05/25 by Yumeng Ou, Ou, Yumeng, Krystal Taylor +1 · 2 citations
Mathematics · #28A #42B10 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2005.12233
openalex publication_date 2020/05/25 · openalex created_date 2020/05/29 · openalex updated_date 2026/07/28
In this article, we study two problems concerning the size of the set of finite point configurations generated by a compact set E⊂ ℝd. 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 (5)/(4), then there exists a point x∈ E so that for each integer k≥2, 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 (7)/(4), we show that the dimension of the set of vertices in E forming said triangle does not exceed 3 dimH (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.