Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets

We study Minkowski contents and fractal curvatures of arbitrary self-similar tilings (constructed on a feasible open set of an IFS) and the general relations to the corresponding functionals for self-similar sets. In particular, we characterize the situation, when these functionals coincide. In this case, the Minkowski content and the fractal curvatures of a self-similar set can be expressed completely in terms of the volume function or curvature data, respectively, of the generator of the tiling. In special cases such formulas have been obtained recently using tube formulas and complex dimensions or as a corollary to results on self-conformal sets. Our approach based on the classical Renewal Theorem is simpler and works for a much larger class of self-similar sets and tilings. In fact, generator type formulas are obtained for essentially all self-similar sets, when suitable volume functions (and curvature functions, respectively) related to the generator are used. We also strengthen known results on the Minkowski measurability of self-similar sets, in particular on the question of non-measurability in the lattice case.Comment: 28 pages, 2 figure

Geometry of canonical self-similar tilings

We give several different geometric characterizations of the situation in which the parallel set $F_\epsilon$ of a self-similar set $F$ can be described by the inner $\epsilon$-parallel set $T_{-\epsilon}$ of the associated canonical tiling $\mathcal T$, in the sense of \cite{SST}. For example, $F_\epsilon=T_{-\epsilon} \cup C_\epsilon$ if and only if the boundary of the convex hull $C$ of $F$ is a subset of $F$, or if the boundary of $E$, the unbounded portion of the complement of $F$, is the boundary of a convex set. In the characterized situation, the tiling allows one to obtain a tube formula for $F$, i.e., an expression for the volume of $F_\epsilon$ as a function of $\epsilon$. On the way, we clarify some geometric properties of canonical tilings. Motivated by the search for tube formulas, we give a generalization of the tiling construction which applies to all self-affine sets $F$ having empty interior and satisfying the open set condition. We also characterize the relation between the parallel sets of $F$ and these tilings.Comment: 20 pages, 6 figure

Lower S-dimension of fractal sets

AbstractThe interrelations between (upper and lower) Minkowski contents and (upper and lower) surface area based contents (S-contents) as well as between their associated dimensions have recently been investigated for general sets in Rd (cf. Rataj and Winter (in press) [6]). While the upper dimensions always coincide and the upper contents are bounded by each other, the bounds obtained in Rataj and Winter (in press) [6] suggest that there is much more flexibility for the lower contents and dimensions. We show that this is indeed the case. There are sets whose lower S-dimension is strictly smaller than their lower Minkowski dimension. More precisely, given two numbers s, m with 0<s<m<1, we construct sets F in Rd with lower S-dimension s+d−1 and lower Minkowski dimension m+d−1. In particular, these sets are used to demonstrate that the inequalities obtained in Rataj and Winter (in press) [6] regarding the general relation of these two dimensions are best possible

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

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 $\mathbb{R}$ by Falconer, Lapidus and v.~Frankenhuijsen, and the forward direction was shown for fractal subsets of $\mathbb{R}^d$, $d \geq 2$, 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.Comment: 20 pages, 6 figure

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

In a previous paper by the first two authors, a tube formula for fractal sprays was obtained which also applies to a certain class of self-similar fractals. The proof of this formula uses distributional techniques and requires fairly strong conditions on the geometry of the tiling (specifically, the inner tube formula for each generator of the fractal spray is required to be polynomial). Now we extend and strengthen the tube formula by removing the conditions on the geometry of the generators, and also by giving a proof which holds pointwise, rather than distributionally. Hence, our results for fractal sprays extend to higher dimensions the pointwise tube formula for (1-dimensional) fractal strings obtained earlier by Lapidus and van Frankenhuijsen. Our pointwise tube formulas are expressed as a sum of the residues of the "tubular zeta function" of the fractal spray in $\mathbb{R}^d$. This sum ranges over the complex dimensions of the spray, that is, over the poles of the geometric zeta function of the underlying fractal string and the integers $0,1,...,d$. The resulting "fractal tube formulas" are applied to the important special case of self-similar tilings, but are also illustrated in other geometrically natural situations. Our tube formulas may also be seen as fractal analogues of the classical Steiner formula.Comment: 43 pages, 13 figures. To appear: Advances in Mathematic

Minkowski measurability results for self-similar tilings and fractals with monophase generators

In a previous paper [arXiv:1006.3807], 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.Comment: 18 pages, 1 figur