Complex Dimensions of Self-Similar Fractal Strings and Diophantine Approximation
Michel Lapidus and Machiel van Frankenhuysen
In "Fractal Geometry and Number Theory" (Complex dimensions of fractal strings and zeros of zeta functions), Research Monograph, Birkhäuser, Boston, 2000, the authors develop a theory of complex dimensions of fractal strings (one-dimensional fractal drums).
In this paper, we discuss the computational aspects of these explicit formulas for self-similar fractal strings. For these strings, the problem of finding the complex dimensions becomes accessible to computation and mathematical experimentation, as well as to detailed theoretical investigation, since it amounts to solving a Dirichlet polynomial equation. We distinguish the lattice case and the nonlattice case. Our results suggest, in particular, that a nonlattice string is fractal in a countable set of fractal dimensions that is dense in a connected interval. We illustrate our theory by means of several examples. In the long run, this work is aimed at developing a Diophantine approximation theory of self-similar strings (and sets), both qualitatively and quantitatively.