Research Problems in Function Theory — Problem 5.48

Definitions and mathematical statement

Let $\mathbb D=\{z:|z|<1\}$. An analytic function with Hadamard gaps is a holomorphic function admitting an expansion \[ f(z)=a_0+\sum_{k\ge1}a_kz^{n_k},\qquad 0<n_1<n_2<\cdots,\quad \frac{n_{k+1}}{n_k}\ge q>1, \] where the exponents are integers and the displayed series converges in $\mathbb D$. One can omit zero coefficients and disregard finitely many initial terms without changing the gap requirement relevant to unboundedness. Characterize the subsets $S\subset\mathbb D$ having the following boundedness-determining property: \[ \sup_{z\in S}|f(z)|<\infty \quad\Longrightarrow\quad \sup_{z\in\mathbb D}|f(z)|<\infty \] for every such Hadamard-gap function $f$. The gap ratio and the exponent sequence may depend on $f$; if a characterization depends on a fixed ratio $q$ or sequence, that dependence should be stated as a separate restricted version. The bound on $f$ over $S$ may depend on the particular function. The question asks whether boundedness propagates, not whether there is a single universal constant comparing the two suprema. No discreteness, closedness, or radial form is assumed for $S$; a useful answer should explain which geometric or sampling properties of $S$ control all these lacunary analytic functions.

Short English statement

Which subsets of the disc detect boundedness of every analytic function with geometrically separated Taylor exponents?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305048 (AMR-022-5048), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 5.48. Original source indexed by AMR-022-5048. https://arxiv.org/abs/1809.07200.