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arXiv · 2511.17414

Perfect Sets of Liouville Numbers with Controlled Self-Powers

Sidney A. Morris·Marcelo O. Ribeiro·Diego Marques
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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 $x\mapsto x^x$.

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Sidney A. Morris, Marcelo O. Ribeiro, Diego Marques. 2025-11-21. Perfect Sets of Liouville Numbers with Controlled Self-Powers. https://arxiv.org/abs/2511.17414

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