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• W1856366937 abstract "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}." @default.
• W1856366937 created "2016-06-24" @default.
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• W1856366937 date "2017-09-01" @default.
• W1856366937 modified "2023-10-18" @default.
• W1856366937 title "Complex dimensions of fractals and meromorphic extensions of fractal zeta functions" @default.
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• W1856366937 doi "https://doi.org/10.1016/j.jmaa.2017.03.059" @default.
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