Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers
Abstract
Let ℒ denote the set of Liouville numbers. We prove a local arithmetic rigidity theorem for real-analytic functions. If U⊂ℝ is an open interval and f:U→ℝ is real-analytic with f(ℒ∩ U)⊆ℒ, then f is the restriction to U of a rational function in ℝ(x). Quantitatively, there exists an absolute constant τ>2 such that, for every nonrational real-analytic function f and every nonempty open subinterval V⊂ U, the set of ξ∈ V∩ℒ satisfying μ(f(ξ))≤τ contains a Cantor set. As a consequence, every entire function F:ℂ→ℂ satisfying F(ℒ)⊆ ℒ is a polynomial with real coefficients. This gives a negative answer to a problem posed by Mahler in 1984. The proof develops a two-height counting estimate for rational approximation to analytic graphs, treating source and target denominators separately, and combines it with a nested construction producing Liouville inputs whose images have uniformly bounded irrationality exponent.
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BibTeX
@misc{marques2026arithmetic,
title = {{Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers}},
author = {Diego Marques},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.14202},
eprint = {2609.14202},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14202},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers
AU - Diego Marques
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.14202
UR - https://arxiv.org/abs/2609.14202
AB - Let ℒ denote the set of Liouville numbers. We prove a local arithmetic rigidity theorem for real-analytic functions. If U⊂ℝ is an open interval and f:U→ℝ is real-analytic with f(ℒ∩ U)⊆ℒ, then f is the restriction to U of a rational function in ℝ(x). Quantitatively, there exists an absolute constant τ>2 such that, for every nonrational real-analytic function f and every nonempty open subinterval V⊂ U, the set of ξ∈ V∩ℒ satisfying μ(f(ξ))≤τ contains a Cantor set. As a consequence, every entire function F:ℂ→ℂ satisfying F(ℒ)⊆ ℒ is a polynomial with real coefficients. This gives a negative answer to a problem posed by Mahler in 1984. The proof develops a two-height counting estimate for rational approximation to analytic graphs, treating source and target denominators separately, and combines it with a nested construction producing Liouville inputs whose images have uniformly bounded irrationality exponent.
ER -