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Proceedings of the London Mathematical Society Advance Access originally published online on October 9, 2008
Proceedings of the London Mathematical Society 2009 98(3):559-584; doi:10.1112/plms/pdn037
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© 2008 London Mathematical Society

A quantitative version of the Besicovitch projection theorem via multiscale analysis

Terence Tao

UCLA
Department of Mathematics
Los Angeles, CA 90095-1596
USA
http://www.math.ucla.edu/~tao

Received 24 August 2007.

By using a multiscale analysis, we establish quantitative versions of the Besicovitch projection theorem (almost every projection of a purely unrectifiable set in the plane of finite length has measure zero) and a standard companion result, namely that any planar set with at least two projections of measure zero is purely unrectifiable. We illustrate these results by providing an explicit (but weak) upper bound on the average projection of the nth generation of a product Cantor set.

2000 Mathematics Subject Classification 28A75


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