PreprintCC BY 4.0arXiv
On Mahler U-numbers under rational maps
Abstract
In 1906, Maillet proved that every nonconstant rational function over ℚ maps Liouville numbers to Liouville numbers. Let R be a nonconstant rational function with real algebraic coefficients, of degree d, and let e be the degree of its minimal field of definition over ℚ. We prove that, for every m≥ 1, the possible types of R(ξ), as ξ ranges over U-numbers of type m, are exactly the integers r with ⌈ m/(ed)⌉≤ r≤ em. Moreover, every such type occurs for some ξ of type m in every nonempty open interval. Consequently, for m≥2, the only rational functions with real algebraic coefficients preserving the class of U-numbers of type m are the Möbius transformations over ℚ.
Related
- Mahler's Problem on Liouville Numbers II. Rational Maps
- Smooth Diffeomorphisms and Mahler's Problem on Liouville Numbers
- Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers
- Rational Points on Danielewski Surfaces over Finite fields and Polygonal Numbers
- Higher arithmetic degrees of dominant rational self-maps
BibTeX
@misc{marques2026mahler,
title = {{On Mahler $U$-numbers under rational maps}},
author = {Diego Marques},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03988},
eprint = {2610.03988},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03988},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Mahler U-numbers under rational maps
AU - Diego Marques
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03988
UR - https://arxiv.org/abs/2610.03988
AB - In 1906, Maillet proved that every nonconstant rational function over ℚ maps Liouville numbers to Liouville numbers. Let R be a nonconstant rational function with real algebraic coefficients, of degree d, and let e be the degree of its minimal field of definition over ℚ. We prove that, for every m≥ 1, the possible types of R(ξ), as ξ ranges over U-numbers of type m, are exactly the integers r with ⌈ m/(ed)⌉≤ r≤ em. Moreover, every such type occurs for some ξ of type m in every nonempty open interval. Consequently, for m≥2, the only rational functions with real algebraic coefficients preserving the class of U-numbers of type m are the Möbius transformations over ℚ.
ER -