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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 2016

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On Intersections of Generic Perturbations of Definable Sets

Mycielski J., Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 2/3

95--103

Consider an o-minimal expansion 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≤ dim B < dim M, then dim(σ(A) ∩ B) < dim A for a generic σ in E; here dim ∅ = −1. And if E acts s.l.t. on M and A and B are two definable subsets of M, then dim(σ(A) ∩ B) ≤ max{dim A + dim B − m, −1} for a generic σ 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 ⊂ 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.

Intersection of Generic Rotations in Some Classical Spaces

Nowak K., Tomkowicz G.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 2/3

105--107

Consider an o-minimal structure on the real field R and two definable subsets A, B of the Euclidean space Rn, of the unit sphere Sn or of the hyperbolic space Hn, n ≥ 2, which are of dimensions k, l ≤ n−1, respectively. We prove that the dimension of the intersection σ(A) ∩ B is less than min{k, l} for a generic rotation σ of the ambient space; here we set dim ∅ = −1.