A fundamental dichotomy for definably complete expansions of ordered fields

TL;DR

This paper establishes a fundamental dichotomy for expansions of definably complete fields, showing they either define a discrete subring or have a certain density property, and applies this to a definable Lebesgue differentiation theorem.

Contribution

It introduces a key dichotomy in definably complete fields and derives a definable Lebesgue differentiation theorem as a consequence.

Findings

Either defines a discrete subring or the image of a discrete set is nowhere dense

Provides a definable version of Lebesgue's differentiation theorem

Clarifies structural properties of definably complete fields

Abstract

An expansion of a definably complete field either defines a discrete subring, or the image of a definable discrete set under a definable map is nowhere dense. As an application we show a definable version of Lebesgue's differentiation theorem.