Question 3.27 — Mahler

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A Liouville number is a real irrational $x$ such that for every integer $N\ge1$ there are integers $p$ and $q\ge2$ with $0<|x-p/q|<q^{-N}$. An entire transcendental function is a holomorphic function on all of $\mathbb C$ which is not a polynomial.

Statement recorded in the supplied source.

Does there exist an entire transcendental function $f$ such that $f(x)$ is a real Liouville number for every real Liouville number $x$? [S2]

Short English statement

Can an entire function which is not a polynomial send every Liouville number to another Liouville number?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700045 (AMR-086-0045). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Question 3.27. https://arxiv.org/abs/math/0312440.