Research Problems in Function Theory — Problem 2.46

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 $\mathfrak c=|\mathbb R|=2^{\aleph_0}$ be the continuum cardinal, let $m$ be an infinite cardinal with $\aleph_0<m<\mathfrak c$, and let $\mathcal A$ be a set of distinct entire functions. Write $m^+$ for the successor cardinal of $m$.

Statement recorded in the supplied source.

If \[\left|\{f(z):f\in\mathcal A\}\right|\le m\quad\text{for every }z\in\mathbb C,\] must $|\mathcal A|\le m$? In particular, determine the case $\mathfrak c=m^+$, or whether its answer is independent of the usual axioms of set theory. [S2] The original paragraph gives the positive answer when $m^+<\mathfrak c$, and motivates the question with Erdős's countable-family result and its continuum-hypothesis counterexample. Update 2.46 reports no further progress. [S2]

Short English statement

If an entire-function family takes at most a specified number of different values at each point, must the family itself have at most that cardinality?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302046 (AMR-022-2046). 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 2.46 and Update 2.46. https://arxiv.org/abs/1809.07200.