Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T18:35:36.168Z Has data issue: false hasContentIssue false

Intersection of Continua and Rectifiable Curves

Published online by Cambridge University Press:  21 August 2013

Richárd Balka
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary, ([email protected]; [email protected])
Viktor Harangi
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary, ([email protected]; [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that for any non-degenerate continuum K ⊆ ℝd there exists a rectifiable curve such that its intersection with K has Hausdorff dimension 1. This answers a question of Kirchheim.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Engelking, R., General topology, revised edn (Heldermann, Berlin, 1989).Google Scholar
2.Falconer, K., Fractal geometry: mathematical foundations and applications, 2nd edn (Wiley, 2003).Google Scholar
3.Gromov, M., Partial differential relations (Springer, 1986).Google Scholar
4.Mattila, P., Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, Volume 44 (Cambridge University Press, 1995).Google Scholar