Intersection of Generic Rotations in Some Classical Spaces

Krzysztof Nowak ¹   ; Grzegorz Tomkowicz ²

¹ Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland
² Centrum Edukacji $G^2$ Moniuszki 9 41-902 Bytom, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 64 (2016) no. 2-3, pp. 105-107

Résumé

Consider an o-minimal structure on the real field $\mathbb {R}$ and two definable subsets $A$, $B$ of the Euclidean space $\mathbb {R}^{n}$, of the unit sphere $\mathbb {S}^{n}$ or of the hyperbolic space $\mathbb {H}^{n}$, $n \geq 2$, which are of dimensions $k,l \leq n-1$, respectively. We prove that the dimension of the intersection $\sigma (A) \cap B$ is less than $\min\{k,l\}$ for a generic rotation $\sigma $ of the ambient space; here we set $\dim\emptyset = -1$.

Détail

DOI : 10.4064/ba8084-10-2016

Keywords: consider o minimal structure real field mathbb definable subsets euclidean space mathbb unit sphere mathbb hyperbolic space mathbb geq which dimensions leq n respectively prove dimension intersection sigma cap min generic rotation sigma ambient space here set dim emptyset