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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.
Philipp Hieronymi. "Expansions of subfields of the real field by a discrete set." Fundamenta Mathematicae 215.2 (2011): 167-175. <http://eudml.org/doc/283210>.
@article{PhilippHieronymi2011, abstract = {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.}, author = {Philipp Hieronymi}, journal = {Fundamenta Mathematicae}, keywords = {defining the set of integers; discrete set; real field; expansions of subfields; cyclic multiplicative subgroups; definably complete expansion; Baire category theorem}, language = {eng}, number = {2}, pages = {167-175}, title = {Expansions of subfields of the real field by a discrete set}, url = {http://eudml.org/doc/283210}, volume = {215}, year = {2011}, }
TY - JOUR AU - Philipp Hieronymi TI - Expansions of subfields of the real field by a discrete set JO - Fundamenta Mathematicae PY - 2011 VL - 215 IS - 2 SP - 167 EP - 175 AB - 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. LA - eng KW - defining the set of integers; discrete set; real field; expansions of subfields; cyclic multiplicative subgroups; definably complete expansion; Baire category theorem UR - http://eudml.org/doc/283210 ER -