Research Problems in Function Theory — Problem 5.67

Definitions and mathematical statement

Let $f$ be analytic in $\mathbb D$ and represented there by a lacunary power series \[ f(z)=\sum_{k=0}^{\infty}a_kz^{n_k},\qquad 1\leq n_0<n_1<\cdots,\qquad n_{k+1}/n_k\geq\lambda>1. \] The exponents are integers, the coefficients are complex, and one fixed gap constant $\lambda$ must work for the entire series. Define its maximum term by $m_0(r,f)=\max_{k\geq0}|a_k|r^{n_k}$ for $0<r<1$, and assume $m_0(r,f)\to\infty$ as $r\uparrow1$.

For angles represented in $[0,2\pi)$, put \[ E_f=\left\{\theta: \liminf_{r\uparrow1} \frac{|f(re^{i\theta})|}{m_0(r,f)}>0\right\}. \] Must $E_f$ have one-dimensional Lebesgue measure zero for every such lacunary series? Equivalently, must almost every radial direction admit radii tending to the boundary along which this normalized modulus tends to zero?

The normalization is the largest individual term, not the maximum modulus of $f$ on a circle or an integral mean. The limit inferior is taken along all real radii approaching $1$; the witnessing subsequence may depend on the angle. The desired assertion imposes no further growth comparison between maximum terms at different radii and no probabilistic assumption on the coefficients. Directions at which the normalized modulus stays bounded away from zero may exist, but the question is whether they can occupy a set of positive angular measure.

Short English statement

For a lacunary analytic series with growing maximum term, is its modulus arbitrarily small relative to that term along almost every radius?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305067 (AMR-022-5067), 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 5.67. Original source indexed by AMR-022-5067. https://arxiv.org/abs/1809.07200.