Transference of density (Q376675)

The authors study the various linear densities of a subset \(E\) of the upper half plane at points of \(\mathbb R\) and the conditions under which these densities are transferable. Starting with an example by Goffman and Sledd they investigate the kind of density transfer that one might expect when \(E\) is a Borel subset of the upper half plane. Although some preliminary results might indicate that \(E\) is Borel would entail the transfer of full density in a given direction to full upper density in some other directions, full density is not transferable to every direction nor is density transferred to every \(x\in\mathbb R\). Furthermore, the authors are concerned with the transfer of density for arbitrary sets. Interesting results involving weak \(0-1\) sets are offered. Note that the existence of weak \(0-1\) sets is independent of the Zermelo-Fraenkel axiom of choice.