On the Complex Dimensions of Nonlattice Fractal Strings in Connection with Dirichlet Polynomials

Abstract

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.

2000 AMS Subject Classification:

• 30Axx
• 28A80
• 11M41

Keywords:

• Dirichlet polynomials
• fractal strings
• complex dimensions of fractal strings
• set of dimensions of fractality
• zeros of entire functions

Acknowledgments

The authors wish to thank the referees for their helpful comments. We are especially grateful to one of the referees for suggesting the comments on the falseness of a conjecture involving the right endpoint of the critical interval.

FUNDING

The second author was partially supported by Vicerrectorado de Investigación, Desarrollo e Innovación de la Universidad de Alicante under project GRE11-23.