Complex Dimensions of Self-Similar Fractal Strings and Diophantine Approximation

Abstract

We study the solutions in s of a "Dirichlet polynomial equation'' $m_1r_1^s+\dots+m_Mr_M^s=1$. We distinguish two cases. In the lattice case, when $r_j=r^{k_j}$ are powers of a common base r, the equation corresponds to a polynomial equation, which is readily solved numerically by using a computer. In the nonlattice case, when some ratio $\log r_j/\log r_1$, $j\geq 2$, is irrational, we obtain information by approximating the equation by lattice equations of higher and higher degree. We show that the set of lattice equations is dense in the set of all equations, and deduce that the roots of a nonlattice Dirichlet polynomial equation have a quasiperiodic structure, which we study in detail both theoretically and numerically.

This question is connected with the study of the complex dimensions of self-similar strings. Our results suggest, in particular, that a nonlattice string possesses a set of complex dimensions with countably many real parts (fractal dimensions) which are dense in a connected interval. Moreover, we find dimension-free regions of nonlattice self-similar strings. We illustrate our theory with several examples.

In the long term, this work is aimed in part at developing a Diophantine approximation theory of (higher-dimensional) self-similar fractals, both qualitatively and quantitatively.