engleski

Gauge Minkowski content and complex dimensions of relative fractal drums

Relative fractal drums or, in short RFDs, are objects which, among else, generalize the notion of a bounded subset of the Euclidean space of arbitrary dimension. We obtain some results connecting the gauge Minkowski content of RFDs (i.e., the generalization of the notion of Minkowski content to the case when the tube function ; that is, the Lebesgue measure of the tubular neighborhood of the RFD, does not satisfy a power-law asymptotics for small values of the argument), to the nature of its complex dimensions. The complex dimensions of the RFD are defined as poles (or more general singularities) of its fractal zeta function and are connected to intrinsic oscillations in the geometry of the RFD. This fact was shown recently by obtaining fractal tube formulas for a large class of RFDs ; that is, by expressing the tube function of the RFD as a sum of residues over the poles of the corresponding (modified) fractal zeta function.

Mellin transform; complex dimensions of a relative fractal drum; relative fractal drum; fractal set; box dimension; fractal zeta function; distance zeta function; tube zeta function; fractal string; Minkowski content; Minkowski measurable set; gauge Minkowski content; gauge function; fractal tube formula; residue; meromorphic extension

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