Big and little Lipschitz one sets

TL;DR

This paper investigates the characterization of sets in real numbers that can be represented as the level sets of big and little Lipschitz functions, introducing new concepts like UDT and strongly one-sided density.

Contribution

It provides a complete characterization of big Lipschitz 1 sets and explores the properties of little Lipschitz 1 sets, introducing new density concepts.

Findings

Every Lipschitz 1 set is G_delta and weakly dense.

Existence of weakly dense G_delta sets that are not Lipschitz 1.

Characterization of Lipschitz 1 sets using UDT and density properties.

Abstract

Given a continuous function $f: {{\mathbb R}}\to {{\mathbb R}}$ we denote the so-called "big Lip" and "little lip" functions by $ {{\mathrm {Lip}}} f$ and $ {{\mathrm {lip}}} f$ respectively}. In this paper we are interested in the following question. Given a set $E {\subset} {{\mathbb R}}$ is it possible to find a continuous function $f$ such that $ {{\mathrm {lip}}} f=\mathbf{1}_E$ or $ {{\mathrm {Lip}}} f=\mathbf{1}_E$? For monotone continuous functions we provide the rather straightforward answer. For arbitrary continuous functions the answer is much more difficult to find. We introduce the concept of uniform density type (UDT) and show that if $E$ is $G_\delta$ and UDT then there exists a continuous function $f$ satisfying $ {{\mathrm {Lip}}} f =\mathbf{1}_E$, that is, $E$ is a $ {{\mathrm {Lip}}} 1$ set. In the other direction we show that every ${{\mathrm {Lip}}} 1$ set is $G_\delta$ and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense $G_{{\delta}}$ sets which are not $ {{\mathrm {Lip}}} 1$. We say that a set $E\subset \mathbb{R}$ is ${{\mathrm {lip}}} 1$ if there is a continuous function $f$ such that ${{\mathrm {lip}}} f=\mathbf{1}_E$. We introduce the concept of strongly one-sided density and show that every ${{\mathrm {lip}}} 1$ set is a strongly one-sided dense $F_\sigma$ set.