Unrectifiable 1-sets have vanishing analytic capacity (Q1271364)

A compact set \(E\) in the complex plane is said to be of zero analytic capacity if each bounded analytic outside \(E\) function vanishes identically. The following conjecture of Vitushkin is proved in this paper: Let \(E\) be a compact set in the complex plane with finite 1-dimensional Hausdorff measure \(H^1(E)\) and let for every rectifiable curve \(L\), \(H^1(E\cap L)=0\). Then \(E\) is of zero analytic capacity.