Intersection of Generic Rotations in Some Classical Spaces
Volume 64 / 2016
Bulletin Polish Acad. Sci. Math. 64 (2016), 105-107
MSC: Primary 03C64; Secondary 51M05, 51M10.
DOI: 10.4064/ba8084-10-2016
Published online: 21 October 2016
Abstract
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 $\dim\emptyset = -1$.
