Expansions of subfields of the real field by a discrete set

Philipp Hieronymi¹

¹ Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL 61801, U.S.A.

Fundamenta Mathematicae, Tome 215 (2011) no. 2, pp. 167-175.

Résumé

Let $K$ be a subfield of the real field, $D\subseteq K$ be a discrete set and $f:D^n \to K$ be such that $f(D^n)$ is somewhere dense. Then $(K,f)$ defines $\mathbb{Z}$. We present several applications of this result. We show that $K$ expanded by predicates for different cyclic multiplicative subgroups defines $\mathbb Z$. Moreover, we prove that every definably complete expansion of a subfield of the real field satisfies an analogue of the Baire category theorem.

Détail

DOI : 10.4064/fm215-2-4

Keywords: subfield real field subseteq discrete set somewhere dense defines mathbb present several applications result expanded predicates different cyclic multiplicative subgroups defines mathbb moreover prove every definably complete expansion subfield real field satisfies analogue baire category theorem