The Hausdorff dimension of metric spaces definable in o-minimal expansions of the real field

TL;DR

This paper proves that the Hausdorff dimension of metric spaces definable in o-minimal structures over the real field is itself a definable function and belongs to the field of powers of the structure, linking geometry and model theory.

Contribution

It establishes that Hausdorff dimension in o-minimal structures is a definable function and resides within the field of powers, extending understanding of geometric measure theory in model-theoretic contexts.

Findings

Hausdorff dimension is an R-definable function

Hausdorff dimension belongs to the field of powers of R

The proof uses topological dichotomy and measure theory in o-minimal structures

Abstract

Let $R$ be an o-minimal expansion of the real field. We show that the Hausdorff dimension of an $R$-definable metric space is an $R$-definable function of the parameters defining the metric space. We also show that the Hausdorff dimension of an $R$-definable metric space is an element of the field of powers of $R$. The proof uses a basic topological dichotomy for definable metric spaces due to the second author, and the work of the first author and Shiota on measure theory over nonarchimedean o-minimal structures.