Are lines much bigger than line segments?

TL;DR

This paper explores the relationship between the Hausdorff dimensions of unions of line segments and full lines in Euclidean space, proposing a conjecture and proving its implications in certain cases, especially in the plane.

Contribution

It introduces a conjecture relating dimensions of unions of segments and lines, and proves it in cases where the dimension of the full line union is at most 2.

Findings

Conjecture (*) implies lower bounds on Hausdorff dimensions of Besicovitch sets.

The conjecture holds in the plane (n=2).

Proven that the conjecture holds if the Hausdorff dimension of B is at most 2.

Abstract

We pose the following conjecture: (*) If A is the union of line segments in 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 R^n has Hausdorff dimension at least n-1 and (upper) Minkowski dimension n. We also prove that conjecture (*) holds if the Hausdorff dimension of B is at most 2, so in particular it holds in the plane.