Uniqueness sets with angular density for spaces of entire functions, III: how to minimize the type

TL;DR

This paper investigates the minimal type measures for uniqueness sets in spaces of entire functions, providing explicit constructions and geometric characterizations for critical cases.

Contribution

It introduces a method to find type-minimizing measures with fewer discrete masses and characterizes the critical uniqueness type geometrically for order two.

Findings

Existence of a type-minimizing measure with less than 2ρ discrete masses.

Explicit geometric description of the critical uniqueness type for ρ=2.

Abstract

This note is the third part of our work devoted to uniqueness sets for spaces of entire functions. Given a discrete set $\Lambda$ with angular density $\Delta$ with respect to the order $\rho$, satisfying some regularity condition, we show that there exists a type-minimizing measure $\Delta_0$ with less than $2\rho$ discrete masses. For the case $\rho=2$, the value of the critical uniqueness type is found in geometric terms.