Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators

Lapidus, Michel L.; Pearse, Erin P. J.; Winter, Steffen

doi:10.48550/arxiv.1006.3807

Cited by 16 publications

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“…as a sum of residues of some associated zeta function which is a generating function of the geometry of the tiling. Tube formulas for fractal sprays have first been obtained in [19][20][21]31], generalizing the tube formulas for fractal strings in [27,28] to higher dimensions. The topic has been pursued in [6-8, 15, 25, 26].…”

Section: Introductionmentioning

confidence: 99%

“…With these results in mind, in both [20,Cor. 8.5] and [21,Rem. 4.4 and §8.4], the authors alluded to the fact that several results concerning Minkowski measurability follow almost immediately from the tube formulas; see also [20,Rem.…”

Section: Introductionmentioning

confidence: 99%

“…More specifically, we give precise geometric conditions, the most restrictive of them being the existence of a polynomial expansion for the inner parallel volume of the generator, under which the lattice-nonlattice dichotomy of Minkowski measurabilty carries over to higher dimensions. The question of Minkowski measurability of self-similar sets and tilings with more general generators will be considered in [22] using the general pointwise tube formulas derived in [21].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we recall the construction of selfsimilar tilings and what it means for a generator of such a tiling to be monophase. In Section 3, we recall those tube formulas from [21] needed for the proof of our main results on Minkowski measurability, which are then formulated and proved in Section 4 for selfsimilar tilings and in Section 5 for the associated self-similar sets.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Minkowski Measurability Results for Self-Similar Tilings and Fractals with Monophase Generators

Lapidus

Pearse

Winter³

2013

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I

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In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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Minkowski Measurability Results for Self-Similar Tilings and Fractals with Monophase Generators

Lapidus

Pearse

Winter³

2013

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I

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Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

Lapidus

Radunović²,

Žubrinić³

2015

Fractal Geometry and Stochastics V

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In 2009, the first author introduced a class of zeta functions, called 'distance zeta functions', which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of 'tube zeta functions', defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of "maximally hyperfractal" compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of R N ).Keywords. Fractal set, fractal zeta functions, distance zeta function, tube zeta function, geometric zeta function of a fractal string, Minkowski content, Minkowski measurability, upper box (or Minkowski) dimension, complex dimensions of a fractal set, holomorphic and meromorphic functions, abscissa of convergence, quasiperiodic function, quasiperiodic set, relative fractal drum, fractal tube formulas.

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Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

2016

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Abstract. A long-standing conjecture of Lapidus claims that under certain conditions, self-similar fractal sets fail to be Minkowski measurable if and only if they are of lattice type. The theorem was established for fractal subsets of R by Falconer, Lapidus and v. Frankenhuijsen, and the forward direction was shown for fractal subsets of R d , d ≥ 2, by Gatzouras. Since then, much effort has been made to prove the converse. In this paper, we prove a partial converse by means of renewal theory. Our proof allows us to recover several previous results in this regard, but is much shorter and extends to a more general setting; several technical conditions appearing in previous versions of this result have now been removed.

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Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators

Cited by 16 publications

References 0 publications

Minkowski Measurability Results for Self-Similar Tilings and Fractals with Monophase Generators

Minkowski Measurability Results for Self-Similar Tilings and Fractals with Monophase Generators

Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

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