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Operator-Related Function Theory and Time-Frequency Analysis pp 117–145Cite as

Recent Progress on Favard Length Estimates for Planar Cantor Sets

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Part of the book series: Abel Symposia ((ABEL,volume 9))

Abstract

The Favard length of a planar set E is the average length of its one-dimensional projections. If E is a purely unrectifiable self-similar set of Hausdorff dimension 1 in the plane, a theorem of Besicovitch asserts that E has Favard length 0. An interesting open question concerns quantitative estimates on the decay of the Favard length of finite iterations of such sets. Such estimates are of interest in geometric measure theory, ergodic theory and complex analysis. We review the recent progress on this question, including work by Nazarov-Peres-Volberg, Bond-Volberg, Laba-Zhai, and Bond-Laba-Volberg.

The author is supported in part by NSERC Discovery Grant 22R80520.

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Notes

  1. 1.

    This was recently solved by Hochman [10].

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Izabella Łaba

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  1. Izabella Łaba
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Correspondence to Izabella Łaba .

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Editors and Affiliations

  1. University of Vienna, Faculty of Mathematics, Vienna, Austria

    Karlheinz Gröchenig

  2. Norwegian University of Science and Technology, Dept of Mathematical Sciences, Trondheim, Norway

    Yurii Lyubarskii

  3. Norwegian University of Science and Technology, Dept of Mathematical Sciences, Trondheim, Norway

    Kristian Seip

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Łaba, I. (2015). Recent Progress on Favard Length Estimates for Planar Cantor Sets. In: Gröchenig, K., Lyubarskii, Y., Seip, K. (eds) Operator-Related Function Theory and Time-Frequency Analysis. Abel Symposia, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-08557-9_5

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