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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 theory of `geometric zeta functions' of bounded fractal strings (initiated also by the first author in the early 1990s). In this memoir, 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 (in a suitable sense) the aforementioned classes of fractal zeta functions. This notion is very general and flexible, enabling us to view practically all of the previously studied aspects of the theory of fractal zeta functions from a unified perspective as well as to go well beyond the previous theory. An unexpected novelty is that the upper box (or Minkowski) dimension associated with an RFD can also assume \em negative values (including infinity), which can be interpreted as a flatness property of the RFD. 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 the question of constructing meromorphic extensions of the distance zeta functions of RFDs, as well as to the construction of transcendentally infinity-quasiperiodic RFDs (i.e., roughly, RFDs with infinitely many quasiperiods, all of which are algebraically independent). We also describe a class of RFDs (and, in particular, a new class of bounded sets), called \em maximal hyperfractals};, such that the critical line of (absolute) 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.

fractal set ; fractal string ; relative fractal drum (RFD) ; fractal zeta functions ; relative distance zeta function ; relative tube zeta function ; geometric zeta function of a fractal string ; relative Minkowski content ; relative Minkowski measurability ; relative upper box (or Minkowski) dimension ; relative complex dimensions of an RFD ; holomorphic and meromorphic functions ; abscissa of absolute and meromorphic convergence ; transcendentally infinity-quasiperiodic function

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