Expansions of the Real Field by Canonical Products

Abstract

AbstractWe 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.