Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces

Abstract

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 van Frankenhuijsen. They are illustrated by several examples and applied to yield a new Minkowski measurability criterion.1 The former statement is briefly justified at the end of this paper, while both statements are explained in detail in [37] and [33, Ch. 5]

Zeta functions and complex dimensions of relative fractal drums: theory, examples and applications

“…We point out that, much as as was the case in the one-dimensional situation in [Lap-vFr3, Chapter 12], based on the general explicit formulas and fractal tube formulas obtained in [Lap-Fr1-3] (see, especially, [Lap-vFr3, Chapters 5 and 8]), the definitions of fractality, critical fractality and (strict) subcritical fractality are justified in part by the general fractal tube formulas obtained in [LapRaŽu5] (see also [LapRaŽu4] and [LapRaŽu1,Chapter 5]). ( 2 ) Indeed, the latter tube formulas show that, under mild assumptions, the presence of nonreal complex dimensions of real part d ∈ R corresponds to oscillations of order d in the geometry of A (or of (A, Ω)).…”

“…We note that much more general tube formulas called "fractal tube formulas" are obtained in [LapRaŽu5] (as well as in [LapRaŽu1,Chapter 5], see also [LapRaŽu4]) for arbitrary bounded ( 3 ) Relative versions are also possible, for example for the RFD (A,…”

Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

“…In closing §7.3, we mention that recently, the first author, Goran Radunovic and DarkoZubrinic have developed a general theory of fractal zeta functions and complex dimensions valid in Euclidean spaces R N of any dimension and for arbitrary bounded subsets of R N (see, e.g., the book [48]). In the process, they have very significantly extended the theory of fractal tube formulas obtained originally for fractal strings in [55,56] and then for higher-dimensional fractal sprays (especially, self-similar sprays) in [43][44][45]; see, especially, [50,51] and [48,Ch. 5].…”

“…5]. Accordingly, it is natural to wonder whether the general theory of fractal zeta functions and fractal tube formulas developed in [48] and [50][51][52][53] can be applied and suitably adapted in order to obtain concrete nonarchimedean tube formulas valid (under appropriate hypotheses) for arbitrary compact subsets of p-adic space (Q p ) N (or more general ultrametric spaces), and, in particular, for arbitrary p-adic self-similar sets in (Q p ) N .…”

“…For more information about the theory of fractal strings (or sprays) and their complex dimensions, beside the books [55], [56], [48] and [32], we refer to [13], [15], [18], [21][22][23][24], [27], [30,31], [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47], [49][50][51][52][53][54], [57], [60], [63], as well as the relevant references therein.…”