Positive analytic capacity but zero Buffon needle probability (Q1107661)

One of Vitushkin's famous questions is whether the analytic capacity \(\gamma\) (\(\cdot)\) and the Buffon length Bu(\(\cdot)\) have the same null sets. (The Buffon length of a planar set E is the average length of the orthogonal projection of E (averaging over all possible directions)). \textit{P. Mattila} proved that the answer to this question is no [Ann. Math., II. Ser. 123, 303-309 (1986; Zbl 0589.28006)]. However, Mattila's argument did not give which of the implications \[ 1)\quad \gamma (E)=0\quad \Rightarrow \quad Bu(E)=0,\quad 2)\quad Bu(E)=0\quad \Rightarrow \quad \gamma (E)=0 \] are false. In this paper an explicit construction is given of a set \(E_ 0\) such that \(\gamma (E_ 0)=1\) and \(Bu(E_ 0)=0\). \(E_ 0\) is a ``crank'', i.e. a limit of certain families of line segments parallel to the x-axis.