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

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