On Intersections of Generic Perturbations of Definable Sets

• Autorzy: Mycielski J. , Nowak K.

Identyfikatory

DOI

10.4064/ba8083-10-2016

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

intersections of definable sets; perturbations of definable sets; locally transitive actions of sets; strongly locally transitive actions of sets; o-minimal structures

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2016
• Tom: Vol. 64, no. 2/3
• Strony: 95--103
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Mycielski J.

• Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, U.S.A.

autor

Nowak K.

• Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

Bibliografia

• [1] L. van den Dries, Tame Topology and O-minimal Structures, Cambridge Univ. Press, 1998.
• [2] M. W. Hirsch, Differential Topology, Springer, New York, 1976.
• [3] S. Łojasiewicz, Introduction to Complex Analytic Geometry, Birkhäuser, Basel, 1991.
• [4] J. Mycielski and G. Tomkowicz, On small subsets in Euclidean spaces, Bull. Polish Acad. Sci. Math., to appear.
• [5] K. J. Nowak, Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings, Ann. Polon. Math. 96 (2009), 247–282.
• [6] K. J. Nowak, A theorem on generic intersections in an o-minimal structure, Fund. Math. 227 (2014), 21–25.
• [7] K. J. Nowak, Quantifier elimination in quasianalytic structures via non-standard analysis, Ann. Polon. Math. 114 (2015), 235–267.
• [8] K. J. Nowak and G. Tomkowicz, Intersection of generic rotations in some classical spaces, Bull. Polish Acad. Sci. Math. 64 (2016), 105–107.
• [9] J.-P. Rolin, P. Speissegger and A. J. Wilkie, Quasianalytic Denjoy–Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), 751–777.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.