A Continuous Transition from $$\mathcal {E}$$-Sets to R-sets and Beyond

Abstract

AbstractThe well-known $$\mathcal {E}$$ E -set introduced by Hayman in 1960 is a countable collection of Euclidean discs in the complex plane, whose subtending angles at the origin have a finite sum. An important special case of an $$\mathcal {E}$$ E -set is known as the R-set, for which the sum of the diameters of the discs is finite. These sets appear in numerous papers in the theories of complex differential and functional equations. A given $$\mathcal {E}$$ E -set (hence an R-set) has the property that the set of angles $$\theta $$ θ for which the ray $$\arg (z)=\theta $$ arg ( z ) = θ meets infinitely many discs in the $$\mathcal {E}$$ E -set has linear measure zero. This paper offers a continuous transition from $$\mathcal {E}$$ E -sets to R-sets and then to much thinner sets. In addition to rays, plane curves that originate from the zero distribution theory of exponential polynomials will be considered. It turns out that almost every such curve meets at most finitely many discs in the collection in question. Analogous discussions are provided in the case of the unit disc $$\mathbb {D}$$ D , where the curves tend to the boundary $$\partial \mathbb {D}$$ ∂ D tangentially or non-tangentially. Finally, these findings will be used for improving well-known estimates for logarithmic derivatives, logarithmic differences and logarithmic q-differences of meromorphic functions, as well as for improving standard results on exceptional sets.