Uniqueness sets with angular density for spaces of entire functions, I. Basics

TL;DR

This work develops a complete framework for uniqueness and zero-set problems for spaces of entire functions with prescribed order $\rho$, focusing on angular (directional) distributions of zeros. Building on Levin-Pfluger theory, it expresses the critical zero type $\sigma_Z(\Lambda)$ and the critical uniqueness type $\sigma_U(\Lambda)$ in terms of a $ ho$-regular angular density $\Delta$ and its associated $ ho$-trigonometrically convex function $h_\Delta$, yielding explicit formulas: for non-integer $\rho$, $\sigma_Z(\Lambda)=\max_t h_\Delta(t)$, and for integer $\rho$, $\sigma_Z(\Lambda)=\min_{k\in T_\rho}\max_t[h_\Delta(t)+k(t)]$, while $\sigma_U(\Lambda)=\inf_{k\in TC_\rho}\max_t[h_\Lambda(t)+k(t)]$. The authors show $\sigma_U(\Lambda)=\sigma_Z(\Lambda)$ for $\rho\le 1/2$ and for $\rho=1$, but provide diverse constructions where inequality holds for other $\rho$, illustrating the richness of the interaction between growth, angular distribution, and multiplier/counter-multiplier phenomena. They also connect these deterministic results to random zero sets in Fock-type spaces, proving almost-sure regularity of randomized sequences and giving critical-density thresholds that determine when the randomization yields zero sets, nonzero sets, or uniqueness sets. The paper culminates with sharp density results for $\mathcal{A}_\rho$ and $\mathcal{B}_\rho$, and a family of explicit examples clarifying when the angular density can attain its extremal values, thereby illuminating how geometric configurations control analytic completeness and uniqueness properties. $

Abstract

This is the first part of our work which is devoted to the uniqueness sets for spaces of entire functions. In this part we consider a set $Λ$ with angular density with respect to the order $ρ>0,$ satisfying the Lindelöf condition. We find the value of the critical zero set type for $Λ$ in geometrical terms. We give a necessary and sufficient condition for the coincidence of the critical zero set type and the critical uniqueness set type. At the end of the paper we present an application of our results to random zero sets in Fock-type spaces.

Figures (3)

• Figure 1: Illustration to Example 2: $y=h_{\Delta_2}(t).$
• Figure 2: Illustration to Example 3: $y=h_{\Delta_3}(t).$
• Figure 3: Illustration to Example 3: $y=h_{\Delta_3}(t)+k(t)$ (bold line)

Theorems & Definitions (19)

• Remark
• Theorem 1.1
• Theorem 1.2
• Corollary 1.1
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8