Some further remarks on good sets

TL;DR

This paper explores properties of good sets in n-fold Cartesian products, demonstrating that certain components need not be full and that bounded geodesic lengths are not essential for bounded solutions in specific cases.

Contribution

It provides new insights into the structure of good sets and the conditions for bounded solutions in higher-dimensional Cartesian products.

Findings

Related components in n-fold products may not be full for n >= 4

Boundedness of geodesic lengths is not necessary for bounded solutions when n >= 4

Results extend understanding of good sets in higher dimensions

Abstract

We show that in n-fold cartesian product, n >= 4, a related component need not be a full component. We also prove that when n >= 4, uniform boundedness of lengths of geodesics is not a necessary condition for boundedness of solutions of (1) for bounded function f.