Mahler's Problem on Liouville Numbers II. Rational Maps
Diego Marques
math.NT
Nov 4, 2026 · v1
TL;DR
The main equivalence (Theorem 1.2), characterizing which real rational functions map Liouville numbers into rationals or Liouville numbers, is formalized in Lean 4 in a public repository.
Abstract
In 1984, Mahler asked which analytic functions map the set $\mathscr L$ 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 $\mathbb L$ of $\mathbb R$ such that, for every nonconstant $R\in\mathbb R(X)$ and every nonempty open interval $I$ in its domain, $R(I\cap\mathscr L)\subseteq\mathscr L$ if and only if $R$ is defined over $\mathbb L$ and $R^{-1}(\mathbb Q)\cap I\cap\mathscr L=\varnothing$. The weaker inclusion $R(I\cap\mathscr L)\subseteq\mathbb Q\cup\mathscr L$ holds if and only if $R$ is defined over $\mathbb L$. If $R$ is not defined over $\mathbb L$, 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 $σ$ for which $σX^m$ preserves $\mathscr L$ for every $m\geq2$, with all nonzero rational fibers consisting of $U_m$-numbers.
Problem
Mahler asked which analytic functions map the set of Liouville numbers into itself. Part I reduced real-analytic solutions to real rational functions. The remaining question is which real rational functions have this property.
Approach
The authors define a canonical subfield L of R through an approximation condition at every scale. Sufficiency is shown by approximating coefficients using the denominators of the Liouville approximations. Necessity uses stable rational interpolation, two-height counting of rational points near the graph at obstructed scales, and a Cantor-set construction. A continued-fraction construction with prescribed prime valuations produces strong Liouville coefficients, and the full equivalence of Theorem 1.2 is formalized in Lean 4.
Results
A nonconstant R maps Liouville numbers in an interval into Q ∪ Liouville numbers if and only if R is defined over L. Exact preservation additionally requires that no rational fiber meets the Liouville numbers in the interval. If R is not defined over L, every interval contains a Cantor set of Liouville numbers whose images have bounded irrationality exponent. The authors also construct continuum many strong Liouville numbers σ for which σX^m preserves Liouville numbers for every m ≥ 2.
