Perfect Sets of Liouville Numbers with Controlled Self-Powers
Abstract
We study the arithmetic behavior of self-powers when 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 .
Cite
@article{arxiv.2511.17414,
title = {Perfect Sets of Liouville Numbers with Controlled Self-Powers},
author = {Sidney A. Morris and Marcelo O. Ribeiro and Diego Marques},
journal= {arXiv preprint arXiv:2511.17414},
year = {2026}
}
Comments
The proof of Theorem 3.1 relies on a false claim: C would be a subset of Liouville numbers, which by Jarn\'ik have Hausdorff dimension 0. For Theorem 2.2 & Proposition 2.1, r_m does not range over a grid of mesh 2/3^m, and the estimates require A_j, U_j arbitrarily large, preventing A_j/B_j, U_j/V_j from converging to x