Entire Functions Mapping Countable Dense Subsets of $\mathbb{R}$ onto Countable Dense Subsets of $\mathbb{C}$

Abstract

In 1977, Karl F. Barth posed the following problem: given countable dense sets $A\subset\mathbb{R}$ and $B\subset\mathbb{C}$, does there exist a transcendental entire function $f$ such that $f(A)=B$ and $f(\mathbb{R}\setminus A)\subset\mathbb{C}\setminus B$? We review results related to this question and prove that there exist transcendental entire functions $f$ such that $f\restriction_A\colon A\to B$ is bijective, $f^{-1}(B)\cap\mathbb{R}=A$, and $f'(a)\neq0$ for every $a\in A$. In fact, the set of such functions has the cardinality of the continuum. At the end, we give two extensions of the result, one for countably many pairwise disjoint pairs of dense sets and one with $\mathbb{R}$ replaced by a closed unbounded subset of $\mathbb{C}$ of planar Lebesgue measure zero.