AI 中文总结
研究给定\(\mathbb{R}\)和\(\mathbb{C}\)的可数稠密子集\(A\)、\(B\)，是否存在超越整函数\(f\)满足特定条件的问题。通过证明存在满足条件的超越整函数，且这类函数集基数为连续统，并给出了结果的两个扩展。
AI 中文摘要
1977年，Karl F. Barth提出问题：给定可数稠密子集\(A\subset\mathbb{R}\)和\(B\subset\mathbb{C}\)，是否存在超越整函数\(f\)使得\(f(A)=B\)且\(f(\mathbb{R}\setminus A)\subset\mathbb{C}\setminus B\)？本文回顾相关结果并证明存在超越整函数\(f\)，使得\(f\restriction_A\colon A\to B\)是双射，\(f^{-1}(B)\cap\mathbb{R}=A\)且对每个\(a\in A\)有\(f'(a)\neq0\)。实际上，这类函数集的基数为连续统。最后给出两个结果扩展。
英文摘要
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.
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