Intersection of Generic Rotations in Some Classical Spaces

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 $\dim\emptyset = -1$.