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ISSN 1088-6826(e) ISSN 0002-9939(p)

The sharp Hausdorff measure condition for length of projections

Author(s): Yuval Peres; Boris Solomyak
Journal: Proc. Amer. Math. Soc. 133 (2005), 3371-3379.
MSC (2000): Primary 28A80; Secondary 28A75, 60D05, 28A78
Posted: 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 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.

References:

1.: L. Carleson, Selected problems on exceptional sets, Van Nostrand, Princeton (1967). MR 0225986 (37:1576)
2.: H. Joyce and P. Mörters, A set with finite curvature and projections of zero length, J. Math. Anal. Appl. 247 (2000), no. 1, 126-135. MR 1766928 (2001j:28006)
3.: R. Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153-155. MR 0248779 (40:2030)
4.: J. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimension, Proc. London Math. Soc. 4 (1954), 257-302. MR 0063439 (16:121g)
5.: P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. MR 333890 (96h:28006)
6.: P. Mattila, Hausdorff dimension, projections, and the Fourier transform, Publ. Math. 48 (2004), 3-48. MR 2044636 (2004k:28018)
7.: Y. Peres and B. Solomyak, How likely is Buffon's needle to fall near a planar Cantor set, Pacific J. Math. 204 (2002), 473-496. MR 1907902 (2003h:28010)

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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: 10.1090/S0002-9939-05-08073-1
PII: S 0002-9939(05)08073-1
Received by editor(s): June 29, 2004
Posted: 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
Copyright of article: Copyright 2005, by Yuval Peres and Boris Solomyak

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