A logarithmic characterization of Arakelian sets

TL;DR

The paper provides a new logarithmic characterization of Arakelian sets by showing that a closed set $F\subset\mathbb{C}$ is Arakelian iff every zero-free function in $A(F)$ admits a holomorphic logarithm on $F$, i.e., there exists $g\in A(F)$ with $e^g=f$ on $F$. The authors connect this to the classical topological conditions defining Arakelian sets (no holes and a bounded union of holes when intersected with enlargements $F\cup K$) and leverage a contradiction argument built around entire functions constructed via Weierstrass factorization. Central to the proof are foundational lemmas on extending logarithmic branches, detecting holomorphic logarithms via winding numbers, and carefully controlling boundary behavior near holes through fillings and analytic curves. The work also outlines extensions to planar domains and Riemann surfaces, suggesting avenues for broader applications in uniform approximation problems beyond the plane.

Abstract

Arakelian's classical approximation theorem \cite{Ar} gives necessary and sufficient conditions such that functions can be uniformly approximated in (unbounded) closed sets $F\subset \mathbb{C}$ by entire functions. The conditions are purely topological and concern the connectedness of the complement of $F$. We give a new characterization of Arakelian sets in terms of logarithmic branches of functions $f\in A(F)$, which are continuous in $F$ and holomorphic in its interior $F^\circ$. Our proof is based on a contradiction argument and the counterexample function that we use is furnished by the Weierstrass factorization theorem.

Theorems & Definitions (22)

• Definition 1
• Definition 2: Arakelian set
• Remark 1
• Remark 2
• Definition 3: Uniform approximation set
• Theorem 1: Arakelian Ar
• Theorem 2
• Theorem 3
• Lemma 1
• proof