Defining the set of integers in expansions of the real field by a closed discrete set
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- by Philipp Hieronymi PDF
- Proc. Amer. Math. Soc. 138 (2010), 2163-2168 Request permission
Abstract:
Let $D\subseteq \mathbb {R}$ be closed and discrete and $f:D^n \to \mathbb {R}$ be such that $f(D^n)$ is somewhere dense. We show that $(\mathbb {R},+,\cdot ,f)$ defines $\mathbb {Z}$. As an application, we get that for every $\alpha ,\beta \in \mathbb {R}_{>0}$ with $\log _{\alpha }(\beta )\notin \mathbb {Q}$, the real field expanded by the two cyclic multiplicative subgroups generated by $\alpha$ and $\beta$ defines $\mathbb {Z}$.References
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Additional Information
- Philipp Hieronymi
- Affiliation: Department of Mathematics & Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada
- Address at time of publication: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
- MR Author ID: 894309
- Email: P@hieronymi.de
- Received by editor(s): July 28, 2009
- Received by editor(s) in revised form: August 20, 2009, September 15, 2009, and October 22, 2009
- Published electronically: February 2, 2010
- Communicated by: Julia Knight
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 2163-2168
- MSC (2010): Primary 03C64; Secondary 14P10
- DOI: https://doi.org/10.1090/S0002-9939-10-10268-8
- MathSciNet review: 2596055