How to avoid a compact set

TL;DR

This paper characterizes when a first-order expansion of the real line can define all compact sets, showing it depends on the coincidence of topological and Hausdorff dimensions on closed definable sets.

Contribution

It establishes a precise condition involving dimension coincidence that determines the definability of all compact sets in expansions of the real line.

Findings

Equivalence between dimension coincidence and definability of all compact sets.

Failure of the same condition when using packing dimension.

Characterization of definability in terms of topological and Hausdorff dimensions.

Abstract

A first-order expansion of the $\mathbb{R}$-vector space structure on $\mathbb{R}$ does not define every compact subset of every $\mathbb{R}^n$ if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if $A \subseteq \mathbb{R}^k$ is closed and the Hausdorff dimension of $A$ exceeds the topological dimension of $A$, then every compact subset of every $\mathbb{R}^n$ can be constructed from $A$ using finitely many boolean operations, cartesian products, and linear operations. The same statement fails when Hausdorff dimension is replaced by packing dimension.