Sets non-thin at \infty in \Bbb C ^m, and the growth of sequences of entire functions of genus zero

Abstract

In this paper we define the notion of non-thin at $\infty$ as follows: Let $E$ be a subset of $\Bbb C^m$. For any $R>0$ define $E_R=E\cap \{z\in \Bbb C ^m :|z|\leq R\}$. We say that $E$ is non-thin at $\infty$ if \lim_{R\to\infty}V_{E_R}(z)=0 for all $z\in \Bbb C^m$, where $V_E$ is the pluricomplex Green function of $E$. This definition of non-thin at $\infty$ has good properties: If $E\subset \Bbb C^m$ is non-thin at $\infty$ and $A$ is pluripolar then $E\backslash A$ is non-thin at $\infty$, if $E\subset \Bbb C^m$ and $F\subset \Bbb C^n$ are closed sets non-thin at $\infty$ then $E\times F\subset \Bbb C^m\times \Bbb C^n$ is non-thin at $\infty$ (see Lemma \ref{Lem1}). Then we explore the properties of non-thin at $\infty$ sets and apply this to extend the results in \cite{mul-yav} and \cite{trong-tuyen}.