An example of an Arakelian glove which is a weak Arakelian set

Abstract

In 1976 at the London Mathematical Symposium on Potential Theory at Durham, England, L. A. Rubel posed the problem of characterising weak Arakelian sets or at least showing whether or not there exists an Arakelian glove which is weak Arakelian. We shall give an example of an Arakelian glove which is weak Arakelian and we shall give a necessary condition for a closed set $F$ in the plane to be weak Arakelian. It seems likely that this necessary condition is also sufficient for $F$ to be weak Arakelian but at the present time we are not able to prove the sufficiency.