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I’ve just uploaded a new paper to the arXiv entitled “A quantitative form of the Besicovitch projection theorem via multiscale analysis“, submitted to the Journal of the London Mathematical Society. In the spirit of my earlier posts on soft and hard analysis, this paper establishes a quantitative version of a well-known theorem in soft analysis, in this case the Besicovitch projection theorem. This theorem asserts that if a subset E of the plane has finite length (in the Hausdorff sense) and is purely unrectifiable (thus its intersection with any Lipschitz graph has zero length), then almost every linear projection E to a line will have zero measure. (In contrast, if E is a rectifiable set of positive length, then it is easy to show that all but at most one linear projection of E will have positive measure, basically thanks to the Rademacher differentiation theorem.)
A concrete special case of this theorem relates to the product Cantor set K, consisting of all points (x,y) in the unit square whose base 4 expansion consists only of 0s and 3s. This is a compact one-dimensional set of finite length, which is purely unrectifiable, and so Besicovitch’s theorem tells us that almost every projection of K has measure zero. (One consequence of this, first observed by Kahane, is that one can construct Kakeya sets in the plane of zero measure by connecting line segments between one Cantor set and another.)