Expansions of subfields of the real field by a discrete set

TL;DR

The paper investigates how expanding subfields of the real numbers with discrete sets and functions can define the integers and explores properties like definable completeness and Baire category analogues.

Contribution

It demonstrates that certain expansions of subfields by discrete sets and functions can define the integers and satisfy a Baire Category Theorem analogue.

Findings

Expansions by discrete sets can define the set of integers.

Predicates for cyclic subgroups lead to defining integers.

Definably complete expansions satisfy a Baire Category Theorem analogue.

Abstract

Let K be a subfield of the real field, D be a discrete subset of K and f : D^n -> K be a function such that f(D^n) is somewhere dense. Then (K,f) defines the set of integers. We present several applications of this result. We show that K expanded by predicates for different cyclic multiplicative subgroups defines the set of integers. Moreover, we prove that every definably complete expansion of a subfield of the real field satisfies an analogue of the Baire Category Theorem.