The sharp Hausdorff measure condition for length of projections

Details The sharp Hausdorff measure condition for length of projections The sharp Hausdorff measure condition for length of projections

2004-06-18

arxiv.org/abs/math/0406375v1

Proc. Amer. Math. Soc. 133 (2005), no. 11, 3371--3379

Mathematics

Classical Analysis and ODEs

11 pages, 1 figure

Scientific paper

In a recent paper, Pertti Mattila asked which gauge functions $\phi$ have the property that for any planar Borel set $A$ with positive Hausdorff measure in gauge $\phi$, the projection of $A$ to almost every line has positive length. We show that integrability near zero of $\phi(r)/(r^2)$, which is known to be sufficient for this property, is also necessary if $\phi$ is regularly varying. Our proof is based on a random construction adapted to the gauge function.

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