Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces. (English) Zbl 1441.11234
Summary: We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the complex dimensions of the compact set under consideration (i.e., over the poles of its fractal zeta function). Our results generalize to higher dimensions (and in a significant way) the corresponding ones previously obtained for fractal strings by the first author and and M. van Frankenhuijsen [Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings. New York, NY: Springer (2006; Zbl 1119.28005)]. They are illustrated by several examples and applied to yield a new Minkowski measurability criterion.
MSC:
| 11M41 | Other Dirichlet series and zeta functions |
| 28A75 | Length, area, volume, other geometric measure theory |
| 28A80 | Fractals |
| 28B15 | Set functions, measures and integrals with values in ordered spaces |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 40A10 | Convergence and divergence of integrals |
Keywords:
Mellin transform; complex dimensions; relative fractal drum; fractal set; box dimension; fractal zeta function; distance zeta function; tube zeta function; fractal string; Minkowski content; Minkowski measurability criterion; Minkowski measurable setCitations:
Zbl 1119.28005References:
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