Intersection of Generic Rotations in Some Classical Spaces

Volume 64 / 2016

Krzysztof Nowak, Grzegorz Tomkowicz 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 $\dim\emptyset = -1$.

Authors

  • Krzysztof NowakInstitute of Mathematics
    Faculty of Mathematics and Computer Science
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland
    e-mail
  • Grzegorz TomkowiczCentrum Edukacji $G^2$
    Moniuszki 9
    41-902 Bytom, Poland
    e-mail

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