Zeta functions and complex dimensions of relative fractal drums: theory, examples and applications
In 2009, the first author introduced a new class of zeta functions, called “distance zeta functions”, associated with arbitrary compact fractal subsets of Euclidean spaces of arbitrary dimension. It represents a natural, but nontrivial extension of the existing theory of “geometric zeta functions” of bounded fractal strings. In this work, we introduce the class of “relative fractal drums” (or RFDs), which contains the classes of bounded fractal strings and of compact fractal subsets of Euclidean spaces as special cases. Furthermore, the associated (relative) distance zeta functions of RFDs extend the aforementioned classes of fractal zeta functions. The abscissa of (absolute) convergence of any relative fractal drum is equal to the relative box dimension of the RFD. We pay particular attention to constructing meromorphic extensions of the distance zeta functions of RFDs, as well as to the construction of transcendentally $\infty$-quasiperiodic RFDs. We also describe a class of RFDs, called maximal hyperfractals, such that the critical line of convergence consists solely of nonremovable singularities of the associated relative distance zeta functions. Finally, we also describe a class of Minkowski measurable RFDs which possess an infinite sequence of complex dimensions of arbitrary multiplicity $m\ge1$, and even an infinite sequence of essential singularities along the critical line.