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Expansions of the Real Field by Canonical Products

Part of: Entire and meromorphic functions, and related topics Model theory

Published online by Cambridge University Press:  04 October 2019

Chris Miller  and
Patrick Speissegger
Chris Miller
Affiliation:
Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, Ohio43210, USA Email: miller@math.osu.edu
Patrick Speissegger
Affiliation:
Department of Mathematics & Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, L8S 4K1 Email: speisseg@math.mcmaster.ca
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Abstract

We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated with sequences such as $(-n^{s})_{n>0}$ (for $s>0$) and $(-s^{n})_{n>0}$ (for $s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known so far: (i) the expansion is o-minimal (that is, definable sets have only finitely many connected components); (ii) every Borel subset of each $\mathbb{R}^{n}$ is definable; (iii) the expansion is interdefinable with a structure of the form $(\mathfrak{R}^{\prime },\unicode[STIX]{x1D6FC}^{\mathbb{Z}})$ where $\unicode[STIX]{x1D6FC}>1$, $\unicode[STIX]{x1D6FC}^{\mathbb{Z}}$ is the set of all integer powers of $\unicode[STIX]{x1D6FC}$, and $\mathfrak{R}^{\prime }$ is o-minimal and defines no irrational power functions.

Keywords

o-minimald-minimalAssouad dimensionWeierstrass productGevrey asymptotics

MSC classification

Primary: 03C64: Model theory of ordered structures; o-minimality
Secondary: 30D10: Representations of entire functions by series and integrals 30D15: Special classes of entire functions and growth estimates 30D60: Quasi-analytic and other classes of functions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 3 , September 2020 , pp. 506 - 521
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

Research of author C. M. partially supported by the Simons Foundation and by the Mathematisches Forschungsinstitut Oberwolfach (MFO). Research of author P. S. supported by NSERC of Canada Discovery Grant RGPIN 261961 and the Zukunftskolleg of the University of Konstanz. Preliminary versions of many of the results herein were announced by Miller at the Fields Institute for Research in Mathematical Sciences (August 2016) and at MFO (April–May 2017).

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