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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

The sharp Hausdorff measure condition for length of projections


Authors: Yuval Peres and Boris Solomyak
Journal: Proc. Amer. Math. Soc. 133 (2005), 3371-3379
MSC (2000): Primary 28A80; Secondary 28A75, 60D05, 28A78
Published electronically: June 20, 2005
MathSciNet review: 2161162
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Abstract: In a recent paper, Pertti Mattila asked which gauge functions $\varphi$ have the property that for any Borel set $A\subset\mathbb{R} ^2$ with Hausdorff measure $\mathcal{H}^\varphi(A)>0$, the projection of $A$ to almost every line has positive length. We show that finiteness of $\int_0^1\frac{\varphi(r)}{r^2} dr$, which is known to be sufficient for this property, is also necessary for regularly varying $\varphi$. Our proof is based on a random construction adapted to the gauge function.


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Additional Information

Yuval Peres
Affiliation: Department of Statistics, University of California, Berkeley, California 94720
Email: peres@stat.berkeley.edu

Boris Solomyak
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195
Email: solomyak@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08073-1
PII: S 0002-9939(05)08073-1
Received by editor(s): June 29, 2004
Published electronically: June 20, 2005
Additional Notes: The research of the first author was partially supported by NSF grants #DMS-0104073 and #DMS-0244479. Part of this work was done while he was visiting Microsoft Research. The research of the second author was supported in part by NSF grant #DMS-0099814
Communicated by: David Preiss
Article copyright: © Copyright 2005 by Yuval Peres and Boris Solomyak