A Quantification of a Besicovitch Non-linear Projection Theorem via Multiscale Analysis
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Date
2022-02
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Springer Science and Business Media LLC
Abstract
The Besicovitch projection theorem states that if a subset E of the plane has finite length in the sense of Hausdorff measure and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every orthogonal projection of E to a line will have zero measure. In other words, the Favard length of a purely unrectifiable 1-set vanishes. In this article, we show that when linear projections are replaced by certain non-linear projections called curve projections, this result remains true. In fact, we go further and use multiscale analysis to prove a quantitative version of this Besicovitch non-linear projection theorem. Roughly speaking, we show that if a subset of the plane has finite length in the sense of Hausdorff and is nearly purely unrectifiable, then its Favard curve length is very small. Our techniques build on those of Tao, who in (Proc Lond Math Soc 98:559–584, 2009) proves a quantification of the original Besicovitch projection theorem.
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This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s12220-021-00793-z
Keywords
besicovitch projection theorem, favard curve length, non-linear projections, multiscale analysis