Fässler, Katrin; Orponen, Tuomas
(2014).
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On restricted families of projections in ℝ3.
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Proceedings of the London Mathematical Society, 109(2), pp. 353-381.
Oxford University Press
10.1112/plms/pdu004

We study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 12 -dimensional sets [Math Processing Error] is typically preserved under one-dimensional families of projections onto lines. We improve the result by an ε , proving that if [Math Processing Error], then the packing dimension of the projections is almost surely at least [Math Processing Error]. For projections onto planes, we obtain a similar bound, with the threshold 12 replaced by 1 . In the special case of self-similar sets [Math Processing Error] without rotations, we obtain a full Marstrand-type projection theorem for 1-parameter families of projections onto lines. The [Math Processing Error] case of the result follows from recent work of M. Hochman, but the [Math Processing Error] part is new: with this assumption, we prove that the projections have positive length almost surely.

## Item Type: |
Journal Article (Original Article) |
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## Division/Institute: |
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics |

## UniBE Contributor: |
Fässler, Katrin |

## Subjects: |
500 Science > 510 Mathematics |

## ISSN: |
0024-6115 |

## Publisher: |
Oxford University Press |

## Language: |
English |

## Submitter: |
Prof. Dr. Zoltan Balogh |

## Date Deposited: |
15 Apr 2015 16:18 |

## Last Modified: |
25 Jul 2017 08:00 |

## Publisher DOI: |
10.1112/plms/pdu004 |

## URI: |
https://boris.unibe.ch/id/eprint/66755 |