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:

() *If is the union of line segments in , and is the union of the corresponding full lines, then the Hausdorff dimensions of and agree.*

We prove that this conjecture would imply that every Besicovitch set (compact set that contains line segments in every direction) in has Hausdorff dimension at least and (upper) Minkowski dimension . We also prove that conjecture () holds if the Hausdorff dimension of is at most , 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

Email:
tamas.keleti@gmail.com

DOI:
https://doi.org/10.1090/proc/12978

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