Projections of planar Cantor sets in potential theory
A pseudo-capacity is an upper semicontinuous function \(\gamma :K(\mathbb{R}^2)\rightarrow [0,\infty)\) that vanishes on finite sets. The authors' main result is that if \(\gamma \) is a pseudo-capacity and \(A, B\) are disjoint \(\sigma \)-compacta in \(S=\mathbb{R}/\pi \mathbb{Z}\), such that \(A\) is countable, then, there exists a planar Cantor set \(C\) so that the projection \[ p_{\theta}(x, y)=\textstyle{1\over 2}(x-x\cos 2\theta-y\sin 2\theta ,y+y\cos 2\theta-x\sin 2\theta) \] of the plane onto the line through the origin perpendicular to \(\theta\in A\) is a finite union of intervals \(\forall \theta\in B\) and \(\gamma [p_{\theta}(C)]=0\) \(\forall \theta\in A\). The importance of this topological theorem is that \(\gamma \) is a fairly arbitrary function. In the particular case in which \(\gamma \) is a capacity, one obtains some results of \textit{M. A. Monterie} [ibid., 247-266 (1997)] reviewed in Zbl 0880.31001 above.
scientific article; zbMATH DE number 1052945