Expansions of o-minimal structures by sparse sets (Q2717641)

The authors investigate o-minimal expansions \({\mathcal R}\) of \((\mathbb{R},<,+)\). They show the following: Let \({\mathcal R}\) be such an expansion, \(E\subseteq \mathbb{R}\) with the property that \(F(E^m)\) has no interior for each function \(f: \mathbb{R}^n\to \mathbb{R}\) which is definable in \({\mathcal R}\). Further on assume that every subset of \(\mathbb{R}\) definable in \(({\mathcal R},E)\) has interior or is nowhere dense. Then every subset of \(\mathbb{R}\) definable in \(({\mathcal R},(S))\) has interior or is nowhere dense where \(S\) ranges over all subsets of \(E^k\) \((k\geq 1)\). In order to prove this result the authors use the cell decomposition theorem of van den Dries and generalize the concept of cells to weak cells.NEWLINENEWLINENEWLINEThey give an example of an expansion of the real field that defines an isomorphic copy of the ordered ring of integers but does not define \(\mathbb{Z}\).