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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dimension of slices through the Sierpinski carpet


Authors: Anthony Manning and Károly Simon
Journal: Trans. Amer. Math. Soc. 365 (2013), 213-250
MSC (2010): Primary 28A80; Secondary 37H15, 37C45, 37B10, 37A30
Published electronically: August 30, 2012
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Abstract: For Lebesgue typical $ (\theta ,a)$, the intersection of the Sierpinski carpet $ F$ with a line $ y=x\tan \theta +a$ has (if non-empty) dimension $ s-1$, where $ s=\log 8/\log 3=\dim _{\rm H}F$. Fix the slope $ \tan \theta \in \mathbb{Q} $. Then we shall show on the one hand that this dimension is strictly less than $ s-1$ for Lebesgue almost every $ a$. On the other hand, for almost every $ a$ according to the angle $ \theta $-projection $ \nu ^\theta $ of the natural measure $ \nu $ on $ F$, this dimension is at least $ s-1$. For any $ \theta $ we find a connection between the box dimension of this intersection and the local dimension of $ \nu ^\theta $ at $ a$.


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Additional Information

Anthony Manning
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: A.Manning@warwick.ac.uk

Károly Simon
Affiliation: Institute of Mathematics, Technical University of Budapest, H-1529 P.O. Box 91, Budapest, Hungary
Email: simon@math.bme.hu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05586-3
PII: S 0002-9947(2012)05586-3
Keywords: Self-similar sets, Hausdorff dimension, dimension of fibres.
Received by editor(s): September 3, 2009
Received by editor(s) in revised form: January 24, 2011
Published electronically: August 30, 2012
Additional Notes: This research was supported by the Royal Society grant 2006/R4-IJP and the research of the second author by the OTKA Foundation #T 71693.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.