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Quantitative visibility estimates for unrectifiable sets in the plane


Authors: M. Bond, I. Łaba and J. Zahl
Journal: Trans. Amer. Math. Soc. 368 (2016), 5475-5513
MSC (2010): Primary 28A80; Secondary 28A75, 28A78, 11K55
DOI: https://doi.org/10.1090/tran/6585
Published electronically: April 23, 2015
MathSciNet review: 3458388
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Abstract: The ``visibility'' of a planar set $ S$ from a point $ a$ is defined as the normalized size of the radial projection of $ S$ from $ a$ to the unit circle centered at $ a$. Simon and Solomyak in 2006 proved that unrectifiable self-similar one-sets are invisible from every point in the plane. We quantify this by giving an upper bound on the visibility of $ \delta $-neighborhoods of such sets. We also prove lower bounds on the visibility of $ \delta $-neighborhoods of more general sets, based in part on Bourgain's discretized sum-product estimates.


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Additional Information

M. Bond
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Email: mothwentbad@gmail.com

I. Łaba
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Email: ilaba@math.ubc.ca

J. Zahl
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: jzahl@mit.edu

DOI: https://doi.org/10.1090/tran/6585
Received by editor(s): June 30, 2013
Received by editor(s) in revised form: July 10, 2014
Published electronically: April 23, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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