Dimension of slices through the Sierpinski carpet
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- by Anthony Manning and Károly Simon PDF
- Trans. Amer. Math. Soc. 365 (2013), 213-250 Request permission
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 $s=\log 8/\log 3=\dim _\textrm {H}F$. 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 $a$.References
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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
- MR Author ID: 250279
- Email: simon@math.bme.hu
- 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.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 213-250
- MSC (2010): Primary 28A80; Secondary 37H15, 37C45, 37B10, 37A30
- DOI: https://doi.org/10.1090/S0002-9947-2012-05586-3
- MathSciNet review: 2984058