Sets non-thin at ∞ in

In this paper we define the notion of non-thin at ∞ as follows: Let E be a subset of C m . For any R > 0 define E R = E ∩ { z ∈ C m : | z | ⩽ R } . We say that E is non-thin at ∞ if lim R → ∞ V E R ( z ) = 0 for all z ∈ C m , where V E is the pluricomplex Green function of E. This definition of non-thinness at ∞ has good properties: If E ⊂ C m is non-thin at ∞ and A is pluripolar then E ∖ A is non-thin at ∞; if E ⊂ C m and F ⊂ C n are arbitrary sets, then E and F are non-thin at ∞ iff E × F ⊂ C m × C n 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].