Summary: A theorem of W. J. Schneider [Proc. Symp. Pure Math. 11, 377-385 (1968; Zbl 0186.39301)] states that there is no harmonic function \(h\) in the plane such that \(r^{-\mu} |h(re^{i\theta}) |\to+ \infty\) for all \(\mu>0\) and for a second category set of \(\theta\). Neither the growth rate nor the category condition can be relaxed. This article surveys various analogues and extensions of Schneider’s result. The role of holomorphic and harmonic approximation theorems in the production of counterexamples is indicated.
Sections 1 and 2 are largely a summary of joint work with M. Goldstein [Complex Variables, Theory Appl. 22, No. 3-4, 267-276 (1993; Zbl 0791.31007)]; Section 4 mainly concerns joint work with C. S. Nelson [Analysis 14, No. 2-3, 127-138 (1994; Zbl 0810.31005)]; Sections 3 and 5 contain brief remarks on hitherto unpublished results of the author. Theorem 1.1: Let \(h\) be harmonic on \(\mathbb{R}^N\), \(N\geq 2\). If there is a second category subset \(E\) of \(S_N\) such that (*) \(h(r\xi)= o(\exp (-r^\mu))\) \((r\to+ \infty)\) for all \(\xi\) in \(E\) and all positive numbers \(\mu\), then \(h\equiv 0\). In relation to Theorem 1, the main interest is perhaps in the construction of counterexamples to show its sharpness with respect to both the category condition and the decay rate; harmonic approximation is important in these constructions. The first example concerns the decay rate. Example 1.2: If \(\mu\) is a fixed positive number, then there exists a non-constant harmonic function \(h\) on \(\mathbb{R}^N\) such that (*) holds for each \(\xi\in S_N\).
For the entire collection see [Zbl 0856.00022].