An infinitely generated self-similar set with positive Lebesgue measure and empty interior

TL;DR

This paper constructs an explicit example of an infinitely generated self-similar set that has positive Lebesgue measure but an empty interior, answering a longstanding open question.

Contribution

It provides the first explicit construction of an infinitely generated self-similar set with positive measure and empty interior, advancing understanding of such fractal sets.

Findings

Explicit example of an infinitely generated self-similar set with positive measure

Demonstrates existence of sets with positive measure and empty interior

Addresses a question posed by Peres and Solomyak

Abstract

Peres and Solomyak asked the question: Do there exist self-similar sets with positive Lebesgue measure and empty interior? This question was answered in the affirmative by Cs\"{o}rnyei et al. They gave a parameterised family of iterated function systems for which almost all of the corresponding self-similar sets satisfied the required properties. They do not however provide an explicit example. Motivated by a desire to construct an explicit example, we in this paper provide an explicit construction of an infinitely generated self-similar set with positive Lebesgue measure and empty interior.