Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): Philipp Hieronymi
Journal: Proc. Amer. Math. Soc. 138 (2010), 2163-2168.
MSC (2010): Primary 03C64; Secondary 14P10
Posted: February 2, 2010
MathSciNet review: 2596055
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Let $D\subseteq \mathbb{R}$ be closed and discrete and $f:D^n \to \mathbb{R}$ be such that 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:

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 http://www.math.ohio-state.edu/ $\sim$ 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

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03C64, 14P10

Retrieve articles in all Journals with MSC (2010): 03C64, 14P10

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: 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
Posted: February 2, 2010
Communicated by: Julia Knight
Copyright of article: Copyright 2010, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|