Intersection Properties of Directional Essential Cluster Sets

Abstract

Jarnik in 1936 proved a remarkable property of directional cluster sets. This result states that for a function \(f\) defined on the open upper half plane to the extended real line, each pair of directional cluster sets intersect at all points on the real line but a countable set of points. An example was constructed to show that the exact analogue of Jarnik’s result fails for directional essential cluster sets. Here we shall establish a certain variant of this analogue for directional essential cluster sets of measurable functions.