An optimal extension of Marstrand's plane-packing theorem (Q1423780)

The aim of this paper is to prove an extension of Marstrand's plane packing theorem [\textit{J. Marstrand}, ``Packing planes in \(R^3\)'', Mathematika 26, 180--183 (1979; Zbl 0413.28008)], namely, it is shown that if \(F\) is a subset of the 2--dimensional unit sphere in \(\mathbb R^{3}\) having Hausdorff dimension strictly greater than 1, and \(E\subset \mathbb{R}^{3}\) is a Borel set such that for each \(e\in F\), \(E\) contains a plane perpendicular to the vector \(e\), then \(E\) has positive \(3\)--dimensional Lebesgue measure. This result is optimal in the sense that the condition on the Hausdorff dimension cannot be relaxed.