A dichotomy for expansions of the real field

TL;DR

This paper establishes a clear dichotomy for expansions of the real field, showing that such expansions either define the set of integers or have all bounded nowhere dense definable sets with Minkowski dimension zero.

Contribution

It introduces a fundamental dichotomy in the structure of real field expansions, linking definability of integers to geometric properties of definable sets.

Findings

Either the set of integers is definable or all bounded nowhere dense definable sets have Minkowski dimension zero.

Provides a new characterization of expansions of the real field based on definability and geometric complexity.

Bridges model theory and geometric measure theory in the context of real field expansions.

Abstract

A dichotomy for expansions of the real field is established: Either the set of integers is definable or every nonempty bounded nowhere dense definable subset of the real numbers has Minkowski dimension zero.