Research Problems in Function Theory — Problem 5.54

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 $\mathbb T=\partial\mathbb D$. Let $f(z)=\sum_{n=0}^{\infty}a_nz^n$ converge in $\mathbb D$, and fix $z_0\in\mathbb T$. Assume \[\sum_{n=0}^{\infty}|\operatorname{Re}a_n|=\infty,\qquad\sum_{n=0}^{\infty}|\operatorname{Im}a_n|=\infty.\] Since $|z_0|=1$, these are exactly the two failures of absolute convergence stipulated in the source. Define \[S=\left\{w\in\mathbb C:\exists(\varepsilon_n)_{n\ge0}\in\{-1,1\}^{\{0,1,\ldots\}},\quad\lim_{N\to\infty}\sum_{n=0}^{N}\varepsilon_na_nz_0^n=w\right\}.\] Only ordinary convergent signed series contribute values to $S$.

Statement recorded in the supplied source.

Do these assumptions imply $S=\mathbb C$? [S2]

Short English statement

Under the two stated failures of absolute convergence, can choices of signs make the boundary series converge to every complex value?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305054 (AMR-022-5054). 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 5.54, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.