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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2511.17414 |
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| _version_ | 1866911544855494656 |
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| author | Morris, Sidney A. Ribeiro, Marcelo O. Marques, Diego |
| author_facet | Morris, Sidney A. Ribeiro, Marcelo O. Marques, Diego |
| contents | 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 $x\mapsto x^x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perfect Sets of Liouville Numbers with Controlled Self-Powers Morris, Sidney A. Ribeiro, Marcelo O. Marques, Diego Number Theory 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 $x\mapsto x^x$. |
| title | Perfect Sets of Liouville Numbers with Controlled Self-Powers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.17414 |