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Geometric bounds for Favard length


Author: Tyler Bongers
Journal: Proc. Amer. Math. Soc. 147 (2019), 1447-1452
MSC (2010): Primary 28A78, 28A80
DOI: https://doi.org/10.1090/proc/14358
Published electronically: December 19, 2018
MathSciNet review: 3910411
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Abstract: Given a set in the plane, the average length of its projections over all directions is called Favard length. This quantity measures the size of a set and is closely related to metric and geometric properties of the set such as rectifiability, Hausdorff dimension, and analytic capacity. In this paper, we develop new geometric techniques for estimating Favard length. We will give a short geometrically motivated proof relating Hausdorff dimension to the decay rate of the Favard length of neighborhoods of a set. We will also show that the sequence of Favard lengths of the generations of a self-similar set is convex; this has direct applications to giving lower bounds on Favard length for various fractal sets.


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Tyler Bongers
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: charlesb@math.msu.edu

DOI: https://doi.org/10.1090/proc/14358
Received by editor(s): April 12, 2018
Published electronically: December 19, 2018
Additional Notes: Research partially supported by NSF grant DMS-1056965.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2018 American Mathematical Society