TY - JOUR
T1 - The sharp hausdorff measure condition for length of projections
AU - Peres, Yuval
AU - Solomyak, Boris
PY - 2005/11
Y1 - 2005/11
N2 - In a recent paper, Pertti Mattila asked which gauge functions φ have the property that for any Borel set A ⊂ ℝ 2 with Hausdorff measure H φ (A) > 0, the projection of A to almost every line has positive length. We show that finiteness of ∫ 0 1 φ(r)/r 2 dr, which is known to be sufficient for this property, is also necessary for regularly varying φ. Our proof is based on a random construction adapted to the gauge function.
AB - In a recent paper, Pertti Mattila asked which gauge functions φ have the property that for any Borel set A ⊂ ℝ 2 with Hausdorff measure H φ (A) > 0, the projection of A to almost every line has positive length. We show that finiteness of ∫ 0 1 φ(r)/r 2 dr, which is known to be sufficient for this property, is also necessary for regularly varying φ. Our proof is based on a random construction adapted to the gauge function.
UR - http://www.scopus.com/inward/record.url?scp=27844487410&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-05-08073-1
DO - 10.1090/S0002-9939-05-08073-1
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AN - SCOPUS:27844487410
SN - 0002-9939
VL - 133
SP - 3371
EP - 3379
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 11
ER -