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PreprintCC BY 4.0arXiv

Mahler's Problem on Liouville Numbers II. Rational Maps

  • Diego Marques
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Abstract

In 1984, Mahler asked which analytic functions map the set โ„’ of Liouville numbers into itself. In the first paper of this series, we proved that every real-analytic solution on an open interval is the restriction of a real rational function. We now identify a canonical subfield ๐•ƒ of โ„ such that, for every nonconstant Rโˆˆโ„(X) and every nonempty open interval I in its domain, R(Iโˆฉโ„’)โŠ†โ„’ if and only if R is defined over ๐•ƒ and Rโˆ’1(โ„š)โˆฉ Iโˆฉโ„’=โˆ…. The weaker inclusion R(Iโˆฉโ„’)โŠ†โ„šโˆชโ„’ holds if and only if R is defined over ๐•ƒ. If R is not defined over ๐•ƒ, every such interval contains a Cantor set of Liouville numbers whose images under R are irrational and have irrationality exponents bounded by a constant depending only on R. We also construct continuum many positive strong Liouville numbers ฯƒ for which ฯƒXm preserves โ„’ for every mโ‰ฅ2, with all nonzero rational fibers consisting of Um-numbers.

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Cite

BibTeXRIS
BibTeX
@misc{marques2026mahlers,
  title = {{Mahler's Problem on Liouville Numbers II. Rational Maps}},
  author = {Diego Marques},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07177},
  eprint = {2611.07177},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07177},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Mahler's Problem on Liouville Numbers II. Rational Maps
AU  - Diego Marques
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.07177
UR  - https://arxiv.org/abs/2611.07177
AB  - In 1984, Mahler asked which analytic functions map the set โ„’ of Liouville numbers into itself. In the first paper of this series, we proved that every real-analytic solution on an open interval is the restriction of a real rational function. We now identify a canonical subfield ๐•ƒ of โ„ such that, for every nonconstant Rโˆˆโ„(X) and every nonempty open interval I in its domain, R(Iโˆฉโ„’)โŠ†โ„’ if and only if R is defined over ๐•ƒ and R^โˆ’1(โ„š)โˆฉ Iโˆฉโ„’=โˆ…. The weaker inclusion R(Iโˆฉโ„’)โŠ†โ„šโˆชโ„’ holds if and only if R is defined over ๐•ƒ. If R is not defined over ๐•ƒ, every such interval contains a Cantor set of Liouville numbers whose images under R are irrational and have irrationality exponents bounded by a constant depending only on R. We also construct continuum many positive strong Liouville numbers ฯƒ for which ฯƒX^m preserves โ„’ for every mโ‰ฅ2, with all nonzero rational fibers consisting of U_m-numbers.
ER  -