Sets non-thin at ∞ in CmCm

In this paper we define the notion of non-thin at ∞ as follows: Let E   be a subset of CmCm. For any R>0R>0 define ER=E∩{z∈Cm:|z|⩽R}. We say that E is non-thin at ∞ iflimR→∞VER(z)=0 for all z∈Cmz∈Cm, where VEVE is the pluricomplex Green function of E  . This definition of non-thinness at ∞ has good properties: If E⊂CmE⊂Cm is non-thin at ∞ and A   is pluripolar then E∖AE∖A is non-thin at ∞; if E⊂CmE⊂Cm and F⊂CnF⊂Cn are arbitrary sets, then E and F   are non-thin at ∞ iff E×F⊂Cm×CnE×F⊂Cm×Cn is non-thin at ∞ (see Lemma 2). The results of this paper extend some results in [J. Muller, A. Yavrian, On polynomials sequences with restricted growth near infinity, Bull. London Math. Soc. 34 (2002) 189–199] and [Dang Duc Trong, Tuyen Trung Truong, The growth at infinity of a sequence of entire functions of bounded orders, Complex Var. Elliptic Equ. 53 (8) (2008) 717–743].