Research Problems in Function Theory — Problem 7.6

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Let $(a_n)_{n\geq0}$ be fixed complex coefficients with $\limsup_{n\to\infty}|a_n|^{1/n}\leq1$ and $\sum_{n=0}^{\infty}|a_n|^2=\infty$. Let $(\varepsilon_n)$ be independent random signs with $\mathbb P(\varepsilon_n=1)=\mathbb P(\varepsilon_n=-1)=1/2$, and define the holomorphic random series \[F(z)=\sum_{n=0}^{\infty}\varepsilon_na_nz^n,\qquad z\in\mathbb D.\] The radius assumption states explicitly that the series in the question is defined on $\mathbb D$.

Statement recorded in the supplied source.

Are each of the following true: $\mathbb P(\overline{F(\mathbb D)}=\mathbb C)=1$; $\mathbb P(F(\mathbb D)=\mathbb C)=1$; and, for every fixed $w\in\mathbb C$, $\mathbb P(w\in F(\mathbb D))=1$? The source's update attributes an affirmative answer to the whole-plane assertion to Nazarov, Sodin and Nishry, which also implies the other two assertions. [S2]

Short English statement

Does every random-sign power series with nonsquare-summable coefficients almost surely have dense, or full, complex range and hit each fixed point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307006 (AMR-022-7006). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 7.6, complete problem and update. https://arxiv.org/abs/1809.07200.