Maximal rays of entire functions (Q1611521)

Let \(f\) be a transcendental entire function and \(E(f)= \{\theta:\log|(re^{i\theta})|\sim \log M(r,f)\) as \(r \to \infty\}\), where \(M(r,f)\) is the maximum modulus function of \(f\). The ray \(\{z= re^{i\theta} : 0<r< \infty\), \(\theta \in E\}\) is called a maximal ray of \(f\). Let \(F(f)= [0,2\pi)\backslash E(f)\). An entire function \(f\) is called small if \(\log M(2r,f) \sim \log M(r,f)\) as \(r\to \infty\). Among other results following theorems are proved in the paper: Theorem. If \(f\) is entire and \(F(f)\) is dense in an interval \([\alpha , \beta ]\) then \(E(f)\) is of Baire Category I and so \(F(f)\) is of Category II in \([\alpha , \beta ]\) and residual. Theorem. Given any pre-assigned countable set \(A\) there exists a small entire function \(f\) with \(E(f)= A\).