Are lines much bigger than line segments?

Keleti, Tamás

doi:10.1090/proc/12978

Are lines much bigger than line segments?
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by Tamás Keleti PDF

Proc. Amer. Math. Soc. 144 (2016), 1535-1541 Request permission

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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