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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Defining the set of integers in expansions of the real field by a closed discrete set


Author: Philipp Hieronymi
Journal: Proc. Amer. Math. Soc. 138 (2010), 2163-2168
MSC (2010): Primary 03C64; Secondary 14P10
Published electronically: February 2, 2010
MathSciNet review: 2596055
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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}$.


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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
Email: P@hieronymi.de

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10268-8
PII: S 0002-9939(10)10268-8
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
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.