On Intersections of Generic Perturbations of Definable Sets
Volume 64 / 2016
Abstract
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 $M$.
