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Expansions of the Real Field by Canonical Products
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 3 / September 2020
- Published online by Cambridge University Press:
- 04 October 2019, pp. 506-521
- Print publication:
- September 2020
-
- Article
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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.
