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Limits of Discrete Energy of Families of Increasing Sets

Hari Sarang Nathan

TL;DR

The paper develops a discrete-energy framework to bound the Hausdorff dimension of a set $E$ from finite samples by relating the discrete energy $J_s(P_n)$ of point sets to the continuous Riesz energy $I_s(\mu)$ of measures supported on $E$. It establishes that when a sequence of point sets is $s$-adaptable and the associated counting measures converge weakly to a measure $\mu$, then $J_s(P_n) \to I_s(\mu)$ and $\dim_{\mathcal{H}}(\mathrm{supp}(\mu)) \ge s$, providing a bridge between discrete samples and fractal dimensions. The work extends to random sampling, proving $\mathbb{E}[J_s(P_n)]=I_s(\mu)$ with finite variance characterized by $I_{2s}(\mu)$, and demonstrates probabilistic laws of large numbers for $J_s(P_n)$ in connection to $I_s(\mu)$. Finally, it leverages these results to Erdős/Falconer-type problems, showing that, under Falconer-type assumptions, $s$-adaptable point families yield nontrivial lower bounds on distance-sets and related incidence problems, with extensions to general kernels $\phi$ and dot-product frameworks, highlighting practical implications for high-dimensional data analysis and geometric-combinatorial questions.

Abstract

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.

Limits of Discrete Energy of Families of Increasing Sets

TL;DR

The paper develops a discrete-energy framework to bound the Hausdorff dimension of a set from finite samples by relating the discrete energy of point sets to the continuous Riesz energy of measures supported on . It establishes that when a sequence of point sets is -adaptable and the associated counting measures converge weakly to a measure , then and , providing a bridge between discrete samples and fractal dimensions. The work extends to random sampling, proving with finite variance characterized by , and demonstrates probabilistic laws of large numbers for in connection to . Finally, it leverages these results to Erdős/Falconer-type problems, showing that, under Falconer-type assumptions, -adaptable point families yield nontrivial lower bounds on distance-sets and related incidence problems, with extensions to general kernels and dot-product frameworks, highlighting practical implications for high-dimensional data analysis and geometric-combinatorial questions.

Abstract

The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, , ``fills in'' a set 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 . We also discuss applications to data science and Erdős/Falconer type problems.

Paper Structure

This paper contains 14 sections, 14 theorems, 140 equations, 2 figures.

Table of Contents

  1. Introduction and Notation
  2. Introduction
  3. Notation
  4. Relationship to Data Science
  5. Connections to Other Fractal Dimensions
  6. Points Which Converge to a Measure
  7. Normalized counting measures with weak star convergence
  8. Measures induced by balls
  9. Randomly Drawn Points
  10. Basic statistics
  11. Laws of large numbers
  12. Connections to Erdős/Falconer type problems
  13. Connecting the Falconer and Erdős Distance Problems
  14. Connecting more general Erdős/Falconer type problems

Key Result

Theorem 2.1

Let $s \in (0, d]$. Let $\{e_k\}$ be a sequence of positive real numbers. Then there is a sequence of points, $\{x_n\} \subset \mathbb{R}^d$ which generates $\{P_n\}$ such that there is an increasing subsequence $\{n_k\}$ with $J_s(P_{n_k}) = e_k$.

Figures (2)

  • Figure 1: A sequence of points filling in the unit square via increasingly fine grids.
  • Figure 2: Two measures, $\gamma_{n - 1}$ and $\gamma_{n}$ on the circle. The thick parts indicate the support of $\beta_{n - 1}$ and $\beta_n$. As we go from $\gamma_{n}$ to $\gamma_{n + 1}$, $A_1$ decreases in measure, $A_2$ correspondingly increases in measure, and the measures of the $B_i$ remain unchanged.

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Remark 1.9
  • Example 1.10
  • ...and 33 more