Growth and asymptotic sets of subharmonic functions. II (Q1281294)

[For Part I, cf. J. Lond. Math. Soc. (2) 58, 656-668 (1998).] In the 1960's MacLane, McMillan and Hornblower examined the boundary behaviour of holomorphic and subharmonic functions in relation to the existence of asymptotic values. In the past decade there has been renewed interest in this question, culminating in work of \textit{J. L. Fernández, J. Heinonen} and \textit{J. G. Llorente} [Proc. Lond. Math. Soc. (3) 73, 404-430 (1996; Zbl 0881.31007)] and the reviewer [\textit{S. Gardiner}, J. Lond. Math. Soc. (2) 57, 668-676 (1998)]. In order to be more precise, let \(u\) be a subharmonic function on the upper half-space \(\mathbb{R}^n\times (0,+\infty)\) such that \(u(x,y)\leq y^{-\alpha}\) for \(0< y< 1\), where \(0<\alpha\leq n\). Further, let \(A'\) be the set of boundary points at which \(u\) tends to a value in \((-\infty,+\infty]\) along some curve. It is a consequence of results in the papers cited above that the Hausdorff dimension of \(A'\) is at least \(n-\alpha\). When \(n=1\), this lower bound is known to be sharp, even for harmonic functions \(u\). However, the higher-dimensional situation is harder to treat. The purpose of this paper is to establish the sharpness of this bound for subharmonic functions when \(n\geq 2\). This is achieved by means of a rather complicated construction.