Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers

Abstract

Let $\mathcal{L}$ denote the set of Liouville numbers. We prove a local arithmetic rigidity theorem for real-analytic functions. If $U\subset \mathbb{R}$ is an open interval and $f:U\to \mathbb{R}$ is real-analytic with $f(\mathcal{L}\cap U)\subseteq \mathcal{L}$, then $f$ is the restriction to $U$ of a rational function in $\mathbb{R}(x)$. Quantitatively, there exists an absolute constant $\tau >2$ such that, for every nonrational real-analytic function $f$ and every nonempty open subinterval $V\subset U$, the set of $\xi \in V\cap \mathcal{L}$ satisfying $\mu (f(\xi ))\le \tau$ contains a Cantor set. As a consequence, every entire function $F:\mathbb{C}\to \mathbb{C}$ satisfying $F(\mathcal{L})\subseteq \mathcal{L}$ 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.

arXiv comment41 pages. Includes a Lean 4 formalization