The sharp hausdorff measure condition for length of projections

Yuval Peres, Boris Solomyak

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)3371-3379
Number of pages9
JournalProceedings of the American Mathematical Society
Volume133
Issue number11
DOIs
StatePublished - Nov 2005
Externally publishedYes

Funding

FundersFunder number
Directorate for Mathematical and Physical Sciences0104073, 0244479, 0099814

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