On planar sets with prescribed packing dimensions of line sections (Q2725004)

Let \(\mathbb{L}\) denote the set of the lines of the plane and let \(\mathbb{D}\) be the set of the directions with the usual topology. The author proves in this paper that for every Borel measurable function \(f:\mathbb{L}\rightarrow[0,1]\) there exists a set \(A\subset{\mathbb R}\) such that for a.e. \(\theta\in\mathbb{D}\) for a.e. line \(l\) of direction \(\theta\) we have NEWLINE\[NEWLINE\dim_{\text{P}}(A\cap l)=f(l),NEWLINE\]NEWLINE where \(\dim_{\text{p}}\) denotes the packing dimension. The result answers positively a conjecture posed by Falconer, Järvenpää and Mattila.