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Sets with arbitrary Hausdorff and packing scales in infinite dimensional Banach spaces

Mathieu Helfter

TL;DR

The paper answers Fan's question by constructing compact metric spaces with arbitrarily prescribed scale invariants in infinite dimensions. It first builds Cantor-type Cantor-product sets in an ultrametric space $E$ where the $\varphi$-Hausdorff and $\psi$-packing measures are finite and positive, tied to an equilibrium state $\mu$. It then shows these constructions embed into any infinite-dimensional Banach space via a quasi-Lipschitz embedding, establishing the existence of compact subspaces whose Hausdorff and packing scales take prescribed values $\alpha$ and $\beta$, with corresponding local scales for a natural measure. Central to the approach are density-analytic links between equilibrium states and measures, and a careful construction of product sets that realize the desired scaling via balancing sequences. The results significantly extend dimension-theoretic flexibility to infinite-dimensional contexts, providing explicit methods to realize arbitrary scales in Banach spaces and enriching the theory of metric invariants in high-dimensional geometry.

Abstract

For every couple of Hausdorff functions $ ψ$ and $\varphi $ verifying some mild assumptions, there exists a compact subset $ K $ of the Baire space such that the $ \varphi$-Hausdorff measure and the $ ψ$-packing measure on $ K$ are both finite and positive. Such examples are then embedded in any infinite dimensional Banach space to answer positively a question of Fan on the existence of metric spaces with arbitrary scales.

Sets with arbitrary Hausdorff and packing scales in infinite dimensional Banach spaces

TL;DR

The paper answers Fan's question by constructing compact metric spaces with arbitrarily prescribed scale invariants in infinite dimensions. It first builds Cantor-type Cantor-product sets in an ultrametric space where the -Hausdorff and -packing measures are finite and positive, tied to an equilibrium state . It then shows these constructions embed into any infinite-dimensional Banach space via a quasi-Lipschitz embedding, establishing the existence of compact subspaces whose Hausdorff and packing scales take prescribed values and , with corresponding local scales for a natural measure. Central to the approach are density-analytic links between equilibrium states and measures, and a careful construction of product sets that realize the desired scaling via balancing sequences. The results significantly extend dimension-theoretic flexibility to infinite-dimensional contexts, providing explicit methods to realize arbitrary scales in Banach spaces and enriching the theory of metric invariants in high-dimensional geometry.

Abstract

For every couple of Hausdorff functions and verifying some mild assumptions, there exists a compact subset of the Baire space such that the -Hausdorff measure and the -packing measure on are both finite and positive. Such examples are then embedded in any infinite dimensional Banach space to answer positively a question of Fan on the existence of metric spaces with arbitrary scales.

Paper Structure

This paper contains 10 sections, 10 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Statements
  3. Cantor sets with prescribed Hausdorff and packing measures
  4. Compact subspaces with arbitrary scales
  5. Invariance of scales and embedding compact products in Banach spaces
  6. Quasi-Lipschitz invariance of scales
  7. Embeddings in Banach spaces: proof of \ref{['main scl']}
  8. Hausdorff and packing measure on compact products
  9. Densities of the equilibrium state
  10. Construction of the compact products and proof of \ref{['main1']}

Key Result

Theorem 1

Let $\varphi , \psi \in \mathbb H$. Assume that there exists a constant $C > 0$ such that for every $\varepsilon > 0$: Then there exists a compact product $K \subset E$ with equilibrium state $\mu$ such that: for every Borel subset $X \subset K$. In particular:

Theorems & Definitions (30)

  • Definition 1.1: Hausdorff functions
  • Definition 1.2
  • Definition 2.1: Compact product and equilibrium state
  • Theorem 1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.2: Scaling
  • Definition 2.3: Hausdorff and packing scales
  • ...and 20 more