Research Problems in Function Theory — Problem 2.48

Definitions and mathematical statement

An entire function is transcendental if it is not a polynomial. First let $A\subset\mathbb R$ and $B\subset\mathbb C$ be arbitrary countable dense sets, dense in their indicated spaces. Must there be a transcendental entire function $f$ satisfying \[ f(A)=B,\qquad f(\mathbb R\setminus A)\subset\mathbb C\setminus B? \] The second requirement excludes extra real preimages of $B$; equivalently $f^{-1}(B)\cap\mathbb R=A$. Surjectivity from $A$ onto $B$ is required, but injectivity on $A$ is not stipulated. The source also asks a rigidity question with $E,F\subset\mathbb R$ countable and dense. Suppose an entire function $g$ is real-valued and strictly monotone on $\mathbb R$, and \[ g(E)=F,\qquad g(\mathbb R\setminus E)=\mathbb R\setminus F. \] Find substantive additional hypotheses forcing a simple form for $g$. In particular, for $E=F=\mathbb Q$, identify conditions implying $g(x)=ax+b$ with $a\in\mathbb Q\setminus\{0\}$ and $b\in\mathbb Q$. The known existence of monotone transcendental maps between arbitrary real countable dense sets does not answer this additional rigidity problem or the first mixed real-to-complex question.

Short English statement

Can arbitrary dense countable real and complex sets be matched by an entire function with exactly the desired real preimages, and when do analogous real maps become rigid?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302048 (AMR-022-2048), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.48. Original source indexed by AMR-022-2048. https://arxiv.org/abs/1809.07200.