Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function ζA(s):=∫Aδd(x,A)s−Ndx, where δ>0 is fixed and d(x,A) denotes the Euclidean distance from x to A, has been introduced by the first author in 2009, extending the definition of the zeta function ζL associated with bounded fractal strings L=(ℓj)j≥1 to arbitrary bounded subsets A of the N-dimensional Euclidean space. The abscissa of Lebesgue (i.e., absolute) convergence D(ζA) coincides with D:=dim‾BA, the upper box (or Minkowski) dimension of A. The (visible) complex dimensions of A are the poles of the meromorphic continuation of the fractal zeta function (i.e., the distance or tube zeta function) of A to a suitable connected neighborhood of the “critical line” {Res=D}. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function |At| as t→0+, where At is the Euclidean t-neighborhood of A. We pay particular attention to a class of Minkowski measurable sets, such that |At|=tN−D(M+O(tγ)) as t→0+, with γ>0, and to a class of Minkowski nonmeasurable sets, such that |At|=tN−D(G(log⁡t−1)+O(tγ)) as t→0+, where G is a nonconstant periodic function and γ>0. In both cases, we show that ζA can be meromorphically extended (at least) to the open right half-plane {Res>D−γ} and determine the corresponding visible complex dimensions. Furthermore, up to a multiplicative constant, the residue of ζA evaluated at s=D is shown to be equal to M (the Minkowski content of A) and to the mean value of G (the average Minkowski content of A), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line {Res=D}. Finally, using an appropriate quasiperiodic version of the above construction, with infinitely many suitably chosen quasiperiods associated with a two-parameter family of generalized Cantor sets, we construct “maximally-hyperfractal” compact subsets of RN, for N≥1 arbitrary. These are compact subsets of RN such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line {Res=D}.