Are lines much bigger than line segments? (Q2790261)

If \(A\) is the union of line segments in \(\mathbb{R}^n\), and \(B\) is the union of the corresponding full lines, can \(B\) be much bigger than \(A\)? According to Lebesgue measure, the answer is affirmative, as a classical example by Nikodym shows. It therefore gains interest to study this question from the point of view of Hausdorff dimension (and other dimensions) -- the aim of this well-written paper.NEWLINENEWLINEThe author poses the following conjecture:NEWLINENEWLINE(*) 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 \(B\) are equal.NEWLINENEWLINEAfter that, he proves a number of interesting results implied by (*) (it is worth adding that his conjecture is closely related to the celebrated Kakeya conjecture). Some are quoted below: {\parindent=6mm \begin{itemize}\item[a)] If (*), then every Besicovitch set (compact set that contains line segments in every direction) in \(\mathbb{R}^n\) has Hausdorff dimension \(n-1\); \item[b)] If (*), then every Besicovitch set in \(\mathbb{R}^n\) has packing and upper Minkowski dimension \(n\).NEWLINENEWLINESome evidence supporting the author's conjecture is provided as well: \item[c)] The conjecture (*) holds in the plane; \item[d)] More generally, (*) holds whenever the Hausdorff dimension of \(A\) is less than 2 or the Hausdorff dimension of \(B\) is at most 2.NEWLINENEWLINE\end{itemize}}