Research Problems in Function Theory — Problem 5.66

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 $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. A Blaschke product is a function \[B(z)=\eta z^m\prod_j\frac{|a_j|}{a_j}\frac{a_j-z}{1-\overline{a_j}z},\qquad |\eta|=1,\quad m\in\{0,1,\ldots\},\quad 0<|a_j|<1,\quad\sum_j(1-|a_j|)<\infty,\] where the product can be finite or infinite and repeated zeros are counted with multiplicity. Call the product infinite when its zero sequence has infinitely many terms. For $w\in\mathbb C$, the fibre is $B^{-1}(\{w\})=\{z\in\mathbb D:B(z)=w\}$, counted as distinct points.

Statement recorded in the supplied source.

For every infinite Blaschke product $B$, is there a number $\delta_B>0$ such that $B^{-1}(\{w\})$ is infinite for every $|w|<\delta_B$? The source update reports Stephenson’s negative answer. [S2]

Short English statement

Must every infinite Blaschke product take every sufficiently small value infinitely many times? The source records a counterexample.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305066 (AMR-022-5066). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 5.66, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.