A continuous transition from $\mathcal{E}$-sets to $R$-sets and beyond

TL;DR

This paper explores a continuous spectrum of sets from $\\mathcal{E}$-sets to $R$-sets and beyond, analyzing their intersection properties with rays and curves, and applying these insights to improve estimates in complex analysis.

Contribution

It introduces a continuous transition framework from $\mathcal{E}$-sets to $R$-sets and thinner sets, extending the understanding of their geometric and analytic properties.

Findings

Almost every curve from zero distribution theory intersects finitely many discs.

Extensions to the unit disc show tangential and non-tangential boundary approaches.

Results lead to improved estimates for logarithmic derivatives and exceptional sets.

Abstract

The well-known $\mathcal{E}$-sets introduced by Hayman in 1960 are collections of Euclidean discs in the complex plane with the following property: The set of angles $\theta$ for which the ray $\arg(z)=\theta$ meets infinitely many discs in a given $\mathcal{E}$-set has linear measure zero. An important special case of an $\mathcal{E}$-set is known as the $R$-set. These sets appear in numerous papers in the theories of complex differential and functional equations. This paper offers a continuous transition from $\mathcal{E}$-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 $\mathbb{D}$, where the curves tend to the boundary $\partial\mathbb{D}$ 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.