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Hausdorff dimension, intersections of projections and exceptional plane sections


Authors: Pertti Mattila and Tuomas Orponen
Journal: Proc. Amer. Math. Soc. 144 (2016), 3419-3430
MSC (2010): Primary 28A75
DOI: https://doi.org/10.1090/proc/12985
Published electronically: January 27, 2016
MathSciNet review: 3503710
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Abstract: This paper contains new results on two classical topics in fractal geometry: projections, and intersections with affine planes. To keep the notation of the abstract simple, we restrict the discussion to the planar cases of our theorems.

Our first main result considers the orthogonal projections of two Borel sets $ A,B \subset \mathbb{R}^{2}$ into one-dimensional subspaces. Under the assumptions $ \dim A \leq 1 < \dim B$ and $ \dim A + \dim B > 2$, we prove that the intersection of the projections $ P_{L}(A)$ and $ P_{L}(B)$ has dimension at least $ \dim A - \epsilon $ for positively many lines $ L$, and for any $ \epsilon > 0$. This is quite sharp: given $ s,t \in [0,2]$ with $ s + t = 2$, we construct compact sets $ A,B \subset \mathbb{R}^{2}$ with $ \dim A = s$ and $ \dim B = t$ such that almost all intersections $ P_{L}(A) \cap P_{L}(B)$ are empty. In case both $ \dim A > 1$ and $ \dim B > 1$, we prove that the intersections $ P_{L}(A) \cap P_{L}(B)$ have positive length for positively many $ L$.

If $ A \subset \mathbb{R}^{2}$ is a Borel set with $ 0 < \mathcal {H}^{s}(A) < \infty $ for some $ s > 1$, it is known that $ A$ is ``visible'' from almost all points $ x \in \mathbb{R}^{2}$ in the sense that $ A$ intersects a positive fraction of all lines passing through $ x$. In fact, a result of Marstrand says that such non-empty intersections typically have dimension $ s - 1$. Our second main result strengthens this by showing that the set of exceptional points $ x \in \mathbb{R}^{2}$, for which Marstrand's assertion fails, has Hausdorff dimension at most one.


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Additional Information

Pertti Mattila
Affiliation: Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland
Email: pertti.mattila@helsinki.fi

Tuomas Orponen
Affiliation: Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland
Email: tuomas.orponen@helsinki.fi

DOI: https://doi.org/10.1090/proc/12985
Keywords: Hausdorff dimension, orthogonal projection, plane section
Received by editor(s): September 29, 2015
Published electronically: January 27, 2016
Additional Notes: Both authors were supported by the Academy of Finland, and the second author in particular through the grant Restricted families of projections and connections to Kakeya type problems.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society