A Continuous Transition from E-Sets to R-sets and Beyond

• Published in: The Journal of geometric analysis Vol. 33; no. 9
• Main Authors: Ding, Jie, Heittokangas, Janne, Wen, Zhi-Tao
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2023; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Curves; Diameters; Differential equations; Differential Geometry; Dynamical Systems and Ergodic Theory; Euclidean geometry; Fourier Analysis; Functional equations; Geometry; Global Analysis and Analysis on Manifolds; Logarithms; Mathematical analysis; Mathematics; Mathematics and Statistics; Meromorphic functions; Polynomials; Sums; Set; Tangential limit; Primary 30A10; Logarithmic difference; Stolz angle; Secondary 30D30; Non-tangential limit; Logarithmic derivative
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-023-01336-4

The well-known 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 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 E -set (hence an R -set) has the property that the set of angles θ for which the ray arg ( z ) = θ meets infinitely many discs in the E -set has linear measure zero. This paper offers a continuous transition from 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 D , where the curves tend to the boundary ∂ 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.