Expansions of subfields of the real field by a discrete set

Tom 215 / 2011

Philipp Hieronymi Fundamenta Mathematicae 215 (2011), 167-175 MSC: Primary 03C64; Secondary 14P10, 54E52. DOI: 10.4064/fm215-2-4


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.


  • Philipp HieronymiDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    1409 W. Green Street
    Urbana, IL 61801, U.S.A.

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