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Are lines much bigger than line segments?

Author: Tamás Keleti
Journal: Proc. Amer. Math. Soc. 144 (2016), 1535-1541
MSC (2010): Primary 28A78
Published electronically: December 15, 2015
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Abstract: We pose the following conjecture:

($ \star $) If $ A$ is the union of line segments in $ \mathbb{R}^n$, and $ B$ is the union of the corresponding full lines, then the Hausdorff dimensions of $ A$ and $ B$ agree.

We prove that this conjecture would imply that every Besicovitch set (compact set that contains line segments in every direction) in $ \mathbb{R}^n$ has Hausdorff dimension at least $ n-1$ and (upper) Minkowski dimension $ n$. We also prove that conjecture ($ \star $) holds if the Hausdorff dimension of $ B$ is at most $ 2$, so in particular it holds in the plane.

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Additional Information

Tamás Keleti
Affiliation: Institute of Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary

Keywords: Hausdorff dimension, lines, union of line segments, Besicovitch set, Nikodym set, Kakeya Conjecture.
Received by editor(s): February 2, 2015
Published electronically: December 15, 2015
Additional Notes: The author was supported by Hungarian Scientific Foundation grant no. 104178. Part of this research was done while the author was a visiting researcher at the Alfréd Rényi Institute of Mathematics, whose hospitality is gratefully acknowledged.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 American Mathematical Society