Research Problems in Function Theory — Problem 5.47

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 $\mathbb T=\partial\mathbb D$. Consider an analytic power series with nonzero terms at strictly increasing positive integral exponents, \[f(z)=a_0+\sum_{j=1}^{\infty}a_jz^{\lambda_j},\qquad a_j\ne0,\qquad\lambda_{j+1}/\lambda_j\ge q>1.\] A radial limit at $\zeta\in\mathbb T$ means a limit of $f(r\zeta)$ as $r\uparrow1$, in the Riemann sphere $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$. An asymptotic value at a single boundary point means a spherical limit $f(\gamma(t))\to w$ along a continuous curve $\gamma:[0,1)\to\mathbb D$ with $\gamma(t)\to\zeta\in\mathbb T$.

Statement recorded in the supplied source.

Must every such Hadamard-gap function have at least one radial limit, finite or infinite? If this is false, must it nevertheless have an asymptotic value along a curve ending at one boundary point? The update reports an asymptotic value $\infty$ when the coefficients are unbounded; this does not settle the radial-limit question. [S2]

Short English statement

Determine whether every analytic function with Hadamard gaps has a radial limit somewhere, or at least an asymptotic value along a path to one boundary point.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305047 (AMR-022-5047). 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 5.47, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.