Area-filling curves

TL;DR

This paper explores the properties and constructions of area-filling curves, including homogeneous and invasive variants, and discusses their relationships and historical context within measure theory.

Contribution

It introduces a variant of Knopp's construction for homogeneous area-filling curves and discusses the existence of invasive curves and their connection to Lance-Thomas and Stromberg-Tseng curves.

Findings

Construction of a homogeneous area-filling curve using a variant of Knopp's method

Discussion on the existence of invasive curves with measure 1

Analysis of the relationship between Lance-Thomas and Stromberg-Tseng curves

Abstract

In this paper we study area-filling curves, i.e. continuous and injective mappings defined on [0,1] whose graph has positive measure. Current literature calls them ``Osgood curves", but their invention is due to H. Lebesgue. Stromberg and Tseng constructed homogeneous area-filling curves and offered an elegant example. We show that an appropriate variant of Knopp's construction attains the same homogeneity result. In Section 4 we discuss briefly the existence of an ``invasive" curve, i.e. a continuous and injective mapping from the half-open interval [0,1[ to the unit square, whose image has measure 1. In the last section we discuss several aspects of the Lance-Thomas curve, connecting it with the other construction due to Stromberg and Tseng.