Constructing Whitney sets via IFS with condensation

TL;DR

This paper introduces a novel method using IFS with condensation to construct Whitney sets and arcs, unifying and extending many previous results in the field of Whitney set theory.

Contribution

It presents a new approach with IFS with condensation for constructing Whitney sets, encompassing most earlier constructions as special cases.

Findings

Constructed Whitney sets using IFS with condensation

Unified previous Whitney set constructions under a single framework

Extended the class of Whitney sets that can be constructed

Abstract

In 1935, Whitney constructed a smooth function for which the Morse-Sard Theorem does not hold. Whitney's construction is closely related to certain compact connected set, which is called Whitney set now. From then on, there are a lot of works on Whitney sets. In this paper, we use IFS with condensation, a notion introduced by Barnsley and Demko in 1985, to construct Whitney arcs and Whitney sets. Our construction includes most early results as special cases.