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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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  • 1. O. Belegradek, B. Zilber, The model theory of the field of reals with a subgroup of the unit circle, J. Lond. Math. Soc. (2) 78 (2008) 563-579. MR 2456892 (2009i:03029)
  • 2. L. van den Dries, The field of reals with a predicate for the powers of two, Manuscripta Math. 54 (1985) 187-195. MR 808687 (87d:03098)
  • 3. L. van den Dries, A. Günaydın, The fields of real and complex numbers with a small multiplicative group, Proc. Lond. Math. Soc. (3) 93 (2006) 43-81. MR 2235481 (2007i:03039)
  • 4. H. Friedman, C. Miller, Expansions of o-minimal structures by sparse sets, Fund. Math. 167 (1) (2001) 55-64. MR 1816817 (2001m:03075)
  • 5. A. Günaydın, Model Theory of Fields with Multiplicative Groups, PhD thesis, University of Illinois at Urbana-Champaign (2008).
  • 6. A. Günaydın, P. Hieronymi, The real field with the rational points of an elliptic curve, preprint, arXiv:0906.0528 (2009).
  • 7. C. Miller, Avoiding the projective hierarchy in expansions of the real field by sequences, Proc. Amer. Math. Soc. 134 (5) (2005) 1483-1493. MR 2199196 (2007h:03065)
  • 8. C. Miller, Tameness in expansions of the real field, Logic Colloquium '01 (Vienna), Lect. Notes Log., 20, Assoc. Symbol. Logic, 2005, 281-316. MR 2143901 (2006j:03049)
  • 9. C. Miller, P. Speissegger, Expansions of the real line by open sets: o-minimality and open cores, Fund. Math. 162 (1999) 193-208. MR 1736360 (2001a:03083)
  • 10. C. Miller, P. Speissegger, A trichotomy for expansions of $ \mathbb{R}_{an}$ by trajectories of analytic planar vector fields, preliminary report, available at$ \sim$miller/trichot.pdf.
  • 11. H. Steinhaus, Sur les distances des points de mesure positive, Fund. Math. 1 (1920) 93-104.
  • 12. M. Tychonievich, Defining additive subgroups of the reals from convex subsets, Proc. Amer. Math. Soc. 137 (2009) 3473-3476. MR 2515416

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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

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.

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