On restricted families of projections in ℝ3

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

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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)


08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Fässler, Katrin


500 Science > 510 Mathematics




Oxford University Press




Prof. Dr. Zoltan Balogh

Date Deposited:

15 Apr 2015 16:18

Last Modified:

25 Jul 2017 08:00

Publisher DOI:




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