In this paper we give a new characterization of the closure of the set of the real parts of the zeros of a particular class of Dirichlet polynomials that is associated with the set of dimensions of fractality of certain fractal strings. We show, for some representative cases of nonlattice Dirichlet polynomials, that the real parts of their zeros are dense in their associated critical intervals, confirming the conjecture and the numerical experiments made by M. Lapidus and M. van Frankenhuysen in several papers.The second author was partially supported by Vicerrectorado de Investigación, Desarrollo e Innovación de la Universidad de Alicante under project GRE11-23

A construction analogous to that of Godefroy–Kalton for metric spaces allows one to embed isometrically, in a canonical way, every quasi-metric space $(X,d)$ in an asymmetric normed space $\mathcal {F}_a(X,d)$ (its quasi-metric free space, also called asy