Research Problems in Function Theory — Problem 2.31

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.

Let $A,B\subset\mathbb C$ be countable dense sets; density means that every nonempty open disc meets the set. For a function $f$, $f^{-1}(B)=\{z:f(z)\in B\}$.

Statement recorded in the supplied source.

For every such pair $A,B$, does there exist an entire function $f$ such that \[f^{-1}(B)=A?\] If the universal assertion is false, determine conditions on $A,B$ under which it holds. [S2] Update 2.31 records Barth and Schneider's affirmative construction. This question is about the inverse-image identity; it does not additionally require $f(A)=B$. [S2]

Short English statement

Can an entire function arrange that membership of its value in one prescribed dense countable set occurs exactly on another prescribed dense countable set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302031 (AMR-022-2031). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.31 and Update 2.31. https://arxiv.org/abs/1809.07200.