Metric dimensions and tameness in expansions of the real field
Philipp Hieronymi, Chris Miller
TL;DR
This paper explores the relationship between definability and dimension theory in expansions of the real field, showing that nondefinability of natural numbers aligns with a dimension equality property.
Contribution
It establishes a new equivalence between nondefinability of natural numbers and the equality of topological and Assouad dimensions in definable sets.
Findings
Nondefinability of natural numbers implies dimension equality.
Dimension equality characterizes nondefinability in these structures.
The results connect model theory with geometric dimension concepts.
Abstract
For first-order expansions of the field of real numbers, nondefinability of the set of natural numbers is equivalent to equality of topological and Assouad dimension on images of closed definable sets under definable continuous maps.
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