Swiss cheeses, rational approximation and universal plane curves

TL;DR

This paper develops a new theory of 'allocation maps' for Swiss cheese sets, enabling the construction of complex examples in rational approximation and uniform algebra theory, including sets homeomorphic to the Sierpinski carpet.

Contribution

Introduction of 'allocation maps' theory to modify and construct Swiss cheese sets, solving existing problems and creating examples with desired topological and algebraic properties.

Findings

Constructed Swiss cheese sets homeomorphic to the Sierpinski carpet.

Produced examples of essential, regular uniform algebras on locally connected sets.

Simplified previous technical approaches in the literature.

Abstract

In this paper we consider the compact plane sets known as Swiss cheese sets, which are a useful source of examples in the theory of uniform algebras and rational approximation. We introduce a notion of 'allocation map' connected with Swiss cheeses, and we develop the theory of such maps. We use this theory to modify examples previously constructed in the literature to solve various problems, in order to obtain examples of Swiss cheese sets homeomorphic to the Sierpinski carpet which solve the same problems. In particular, this allows us to give examples of essential, regular uniform algebras on locally connected, compact plane sets. Our techniques also allow us to avoid certain technical difficulties in the literature.