Limits of Discrete Energy of Families of Increasing Sets
This addresses theoretical limits in geometric measure theory and potential applications in data analysis, but appears incremental as it builds on known energy methods.
The paper investigates whether discrete Riesz energy of point sequences approximating a set can bound its Hausdorff dimension, with applications to data science and Erdős/Falconer problems.
The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, $\{x_n\}$, ``fills in'' a set $E \subset \mathbb{R}^d$ in an appropriate sense and investigate the degree to which the discrete analog to the Riesz energy of these sets can be used to bound the Hausdorff dimension of $E$. We also discuss applications to data science and Erdős/Falconer type problems.