Perfect Sets of Liouville Numbers with Controlled Self-Powers

TL;DR

This paper explores the complex behavior of Liouville numbers under self-power operations, constructing a large set where these numbers and their algebraic combinations remain Liouville, revealing intricate algebraic and topological structures.

Contribution

It introduces a novel construction of a perfect set of Liouville numbers with controlled self-powers and algebraic operations, advancing understanding of their arithmetic and topological properties.

Findings

Constructed a perfect set of continuum cardinality of Liouville numbers.

All finite sums, products, and self-powers of these numbers are also Liouville.

Demonstrated rich algebraic and topological structures within the Liouville numbers.

Abstract

We study the arithmetic behavior of self-powers $x^x$ when $x$ is a Liouville number. Using recent ideas on strengthened Liouville approximation, we develop flexible constructions that illuminate how transcendence, Liouville properties, and "large" topological size interact in this setting. As a concrete outcome, we build a perfect set of Liouville numbers of continuum cardinality whose finite sums, finite products, and self-powers all remain Liouville. These results show that rich algebraic and topological structures persist inside the Liouville universe for the map $x\mapsto x^x$.