On Intersections of Generic Perturbations of Definable Sets

Jan Mycielski¹ ; Krzysztof Nowak²

¹ Department of Mathematics University of Colorado Boulder, CO 80309-0395, U.S.A.
² Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 64 (2016) no. 2-3, pp. 95-103.

Résumé

Consider an o-minimal expansion $\mathcal {R}$ of a real closed field $R$ and two definable sets $E$ and $M$. We introduce concepts of a locally transitive (abbreviated to l.t.) and a strongly locally transitive (abbreviated to s.l.t.) action of $E$ on $M$. In the former case, $M$ is supposed to be of pure dimension $m$; in the latter, both $M$ and $E$ are supposed to be of pure dimension. We treat the elements of $E$ as perturbations of the set $M$. We prove that if $E$ acts l.t. on $M$, and $A$ and $B$ are two non-empty definable subsets of $M$ of dimension $\dim A \leq \dim B \lt \dim M$, then $$ \dim (\sigma (A) \cap B) \lt \dim A $$ for a generic $\sigma $ in $E$; here $\dim \emptyset = -1$. And if $E$ acts s.l.t. on $M$ and $A$ and $B$ are two definable subsets of $M$, then $$ \dim (\sigma (A) \cap B) \leq \max\{ \dim A + \dim B -m,-1 \} $$ for a generic $\sigma $ in $E$. We give an example of a l.t. action $E$ on $M$ for which the latter conclusion of the intersection theorem fails. We also prove a theorem on the intersections of generic perturbations in terms of the exceptional set $T \subset M$ of points at which $E$ is not l.t. Finally, we provide some natural conditions which imply that $T$ is a nowhere dense subset of $M$.

Détail

DOI : 10.4064/ba8083-10-2016

Keywords: consider o minimal expansion mathcal real closed field definable sets introduce concepts locally transitive abbreviated strongly locally transitive abbreviated action former supposed pure dimension latter supposed pure dimension treat elements perturbations set prove acts and non empty definable subsets dimension dim leq dim dim dim sigma cap dim generic sigma here dim emptyset acts and definable subsets dim sigma cap leq max dim dim m generic sigma example action which latter conclusion intersection theorem fails prove theorem intersections generic perturbations terms exceptional set subset points which finally provide natural conditions which imply nowhere dense subset