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

Hieronymi, Philipp

doi:10.1090/S0002-9939-10-10268-8

Defining the set of integers in expansions of the real field by a closed discrete set
HTML articles powered by AMS MathViewer

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

Oleg Belegradek and Boris Zilber, The model theory of the field of reals with a subgroup of the unit circle, J. Lond. Math. Soc. (2) 78 (2008), no. 3, 563–579. MR 2456892, DOI 10.1112/jlms/jdn037
Lou van den Dries, The field of reals with a predicate for the powers of two, Manuscripta Math. 54 (1985), no. 1-2, 187–195. MR 808687, DOI 10.1007/BF01171706
Lou van den Dries and Ayhan Günaydın, The fields of real and complex numbers with a small multiplicative group, Proc. London Math. Soc. (3) 93 (2006), no. 1, 43–81. MR 2235481, DOI 10.1017/S0024611506015747
Harvey Friedman and Chris Miller, Expansions of o-minimal structures by sparse sets, Fund. Math. 167 (2001), no. 1, 55–64. MR 1816817, DOI 10.4064/fm167-1-4
A. Günaydın, Model Theory of Fields with Multiplicative Groups, PhD thesis, University of Illinois at Urbana-Champaign (2008).
A. Günaydın, P. Hieronymi, The real field with the rational points of an elliptic curve, preprint, arXiv:0906.0528 (2009).
Chris Miller, Avoiding the projective hierarchy in expansions of the real field by sequences, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1483–1493. MR 2199196, DOI 10.1090/S0002-9939-05-08112-8
Chris Miller, Tameness in expansions of the real field, Logic Colloquium ’01, Lect. Notes Log., vol. 20, Assoc. Symbol. Logic, Urbana, IL, 2005, pp. 281–316. MR 2143901
Chris Miller and Patrick Speissegger, Expansions of the real line by open sets: o-minimality and open cores, Fund. Math. 162 (1999), no. 3, 193–208. MR 1736360
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.
H. Steinhaus, Sur les distances des points de mesure positive, Fund. Math. 1 (1920) 93-104.
Michael A. Tychonievich, Defining additive subgroups of the reals from convex subsets, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3473–3476. MR 2515416, DOI 10.1090/S0002-9939-09-09914-6