| Abstract: |
We study the relation between the growth of a subharmonic functionin the half space Rn+1 + and the size of its asymptotic set. In particular, we prove that for any n ¸ 1 and 0 < ® · n, there exists a subharmonic function u in the Rn+1 + satisfying the growth condition of order ® : u(x) · x¡® n+1 for 0 < xn+1 < 1, such that the Hausdor® dimension of the asymptotic set S ¸6=¡1 A(¸) is exactly n¡®. Here A(¸) is the set of boundary points at which f tends to ¸ along some curve. This proves the sharpness of a theorem due to Berman, Barth, Rippon, Sons, Fern¶andez, Heinonen, Llorente and Gardiner cumulatively. |
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