Uma busca por estruturas lineares e algébricas: um estudo das séries de Dirichlet com faixa de Bohr maximal e das funções holomorfas com cluster grande

Abstract

In this work we investigate the existence of linear and algebraic structures in sets of functions with two distinct singular properties. The first property has to do with the half-planes of convergence (point, uniform or absolute) of Dirichlet series. More precisely: (i) the sets of Dirichlet series with maximal Bohr’s strip; (ii) the set N (resp. the set L) of Dirichlet series whose width of the strip where they converge, but do not converge uniformly (resp. do not converge absolutely), is maximum equal to 1. The second property has to do with cluster set of functions. More specifically: (i) the set of bounded holomorphic functions that have a large (total) cluster (resp. radial cluster) at all possible points; (ii) the set of bounded Dirichlet series that have large cluster sets at every possible points; (iii) the set of bounded holomorphic functions whose linear cluster sets have the cardinality of the continuum at every possible points of the one-dimensional sphere.