On the radial limits of functions with Hadamard gaps (Q1061260)

Let f be a function defined in the unit disk D by the power series \(f(z)=\sum^{\infty}_{k=0}a_ kz^{n_ k}\) where \((n_{k+1})/n_ k\geq \lambda >1\) for \(k=0,1,2,... \). Four theorems and some open problems are discussed concerning the radial limits of f. Among them is the following theorem which generalizes a theorem of \textit{G. Csordas}, \textit{A. J. Lohwater} and \textit{T. Ramsey} [Mich. Math. J. 29, 281-288 (1982; Zbl 0508.30030)]: For \(\lambda >1\), there are positive numbers \(\alpha\), \(\beta\) and \(\gamma\) such that if \(\sum_{k}| a_ k| =\infty\) and \(| a_ k| /(| a_ 0| +| a_ 1| +...+| a_ k|)\leq \alpha,\) (k\(\geq \ell)\), then there is a closed set E in \(\partial D\) with Hausdorff dimension \(\geq \beta\) such that \(Re f(r\zeta)\geq \gamma M(r)\) for \(r_ 0\leq r<1\), \(\zeta\in E\). An example shows, some condition on the \(\{a_ k\}\) or on \(\lambda\) is necessary to get the theorem's conclusion. Two of the theorems are based on work of \textit{J.-P. Kahane} and \textit{G. Weiss} [Ark. Mat. 5, 1-26 (1963; Zbl 0134.057)].