Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On measure estimates arising from Hausdorff distance convergence

Lior Tenenbaum

TL;DR

The paper analyzes when measure convergence can be inferred from Hausdorff convergence of compact sets via fattenings, proving that μ(A_n^{(δ_n)}) → μ(A_∞) under reasonable conditions. It then leverages this result to develop practical spectral-estimation techniques for operator spectra approximated by periodic systems, including corollaries that express spectral measures as limits of measures of expanded spectral bands and propositions that enable numerically stable, Pi1-type computations using Bloch–Floquet theory. The approach unifies set-theoretic limits with measure-theoretic behavior to yield both theoretical guarantees and computational tools for spectral analysis, including dimension bounds in certain cases. This provides a robust framework for estimating spectral measures and dimensions from periodic approximants in high-dimensional discrete Schrödinger operators.

Abstract

We discuss a method to estimate the measure of a compact set which is approximated using the Hausdorff distance by a sequence of compact sets. We do this by considering corresponding fattenings of the sequence of compact sets and showing their measures converge. We further review applications of this result to study the measure of a spectrum of an operator which has a sequence of periodic approximations.

On measure estimates arising from Hausdorff distance convergence

TL;DR

The paper analyzes when measure convergence can be inferred from Hausdorff convergence of compact sets via fattenings, proving that μ(A_n^{(δ_n)}) → μ(A_∞) under reasonable conditions. It then leverages this result to develop practical spectral-estimation techniques for operator spectra approximated by periodic systems, including corollaries that express spectral measures as limits of measures of expanded spectral bands and propositions that enable numerically stable, Pi1-type computations using Bloch–Floquet theory. The approach unifies set-theoretic limits with measure-theoretic behavior to yield both theoretical guarantees and computational tools for spectral analysis, including dimension bounds in certain cases. This provides a robust framework for estimating spectral measures and dimensions from periodic approximants in high-dimensional discrete Schrödinger operators.

Abstract

We discuss a method to estimate the measure of a compact set which is approximated using the Hausdorff distance by a sequence of compact sets. We do this by considering corresponding fattenings of the sequence of compact sets and showing their measures converge. We further review applications of this result to study the measure of a spectrum of an operator which has a sequence of periodic approximations.

Paper Structure

This paper contains 3 sections, 10 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. Estimates for approximations using periodic Schrödinger operators

Key Result

Theorem 1.1

Let $(X,d)$ be a proper metric space, and let $\mu$ be a locally finite Borel measure on $X$. Suppose that $\{A_n\}_{n=1}^\infty\subseteq \mathcal{K}(X)$ is a sequence of compact sets converging to $A_\infty \in \mathcal{K}(X)$ in the Hausdorff distance. Then one has

Theorems & Definitions (21)

  • Example 1.1
  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Remark
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 11 more