Let K be a subfield of the real field, D ⊆ K be a discrete set and f: Dⁿ → K be such that f(Dⁿ) is somewhere dense. Then (K,f) defines ℤ. We present several applications of this result. We show that K expanded by predicates for different cyclic multiplicative subgroups defines ℤ. Moreover, we prove that every definably complete expansion of a subfield of the real field satisfies an analogue of the Baire category theorem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-2-4,
author = {Philipp Hieronymi},
title = {Expansions of subfields of the real field by a discrete set},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {167-175},
zbl = {1270.03059},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-2-4}
}
Philipp Hieronymi. Expansions of subfields of the real field by a discrete set. Fundamenta Mathematicae, Tome 215 (2011) pp. 167-175. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-2-4/