Growth and asymptotic sets of subharmonic functions II

We study the relation between the growth of a subharmonic function in the half space $\Bbb R_+^{n+1}$ and the size of its asymptotic set. In particular, we prove that for any $n \ge 1$ and $0 < \alpha \le n$, there exists a subharmonic function $u$ in the $\Bbb R^{n+1}_+$ satisfying the growth condition of order $\alpha: u(x) \le x_{n+1}^{-\alpha}$ for $0 < x_{n+1} < 1$, such that the Hausdorff dimension of the asymptotic set $\bigcup_{\lambda\ne -\infty} A(\lambda)$ is exactly $n - \alpha$. Here $A (\lambda)$ is the set of boundary points at which $f$ tends to $\lambda$ along some curve. This proves the sharpness of a theorem due to Berman, Barth, Rippon, Sons, Fernández, Heinonen, Llorente and Gardiner cumulatively.