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

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.


Introduction
We begin by stating some definitions and results from the research monograph [8] that will be needed in this article, as well as by recalling some well-known notions. Given a bounded subset A of R N (always assumed to be nonempty in this paper), we denote its δ-neighborhood by A δ := {x ∈ R N : d(x, A) < δ}. Here, d(x, A) := inf{|x − y| : y ∈ A} is the Euclidean distance between the point x and the set A. 1 Furthermore, for a compact subset A of R N and r ≥ 0, we define its upper r-dimensional Minkowski content, M r (A) = lim sup t→0 + t r−N |A t |, and its upper box dimension, dim B A = inf{r ≥ 0 : M r (A) = 0}. The value M r (A) of the lower r-dimensional Minkowski content of A, is defined analogously as M r (A), except for a lower instead of an upper limit, and similarly for the lower box dimension dim B A. If dim B A = dim B A, this common value is called the Minkowski (or box) dimension of A and denoted by dim B A. If 0 < M D (A)(≤)M D (A) < ∞, for some D ≥ 0, the set A is said to be Minkowski nondegenerate. It then follows that dim B A exists and is equal to D. Moreover, if M D (A) exists and is different from 0 and ∞ (in which case dim B A exists and then necessarily, D = dim B A), the set A is said to be Minkowski measurable.
We will now introduce the notions of distance and tube zeta functions of compact sets and state their basic properties. These definitions have enabled us in [8][9][10] to develop a higher-dimensional extension of the theory of complex dimensions of fractal strings ( [12]), valid for arbitrary compact sets.
Definition 1.1 (Fractal zeta functions, [8]). Let A be a compact subset of R N and fix δ > 0. We define the distance zeta function ζ A of A and the tube zeta function ζ A of A by the following Lebesgue integrals, respectively, for some δ > 0 and for all s ∈ C with Re s sufficiently large: It is not difficult to show that the distance and tube zeta functions of a compact subset A of R N satisfy the following functional equation, which is valid on any connected open set U ⊆ C to which any of the two zeta functions has a meromorphic continuation (see [8, §2.2]): Furthermore, in the above definition (see Eq. (1)), the dependence of the zeta functions on the parameter δ > 0 is inessential, from the point of view of the theory of complex dimensions (see Def. 1.4 below). Indeed, it is shown in [8] that the difference of two distance (or tube) zeta functions of the same compact set A, and corresponding to any two different values of the parameter δ, is an entire function.
Let us briefly summarize the main properties of the distance and tube zeta functions (see [8,Ch. 2]): If A is a compact subset of R N , then the tube zeta function ζ A ( · ; δ) is holomorphic in the half-plane {Re s > dim B A} and dim B A coincides with the abscissa of (absolute) convergence of ζ A ( · ; δ). Furthermore, if the box (or Minkowski) dimension D := dim B A exists and M D (A) > 0, then ζ A (s; δ) → +∞ as s ∈ R converges to D from the right. The above statements are also true if we replace ζ A by ζ A and in the preceding sentence assume, in addition, that D < N .
If A is a Minkowski nondegenerate subset of R N (so that D := dim B A exists), and for some δ > 0 there exists a meromorphic extension of ζ A ( · ; δ) to a neighborhood of D, then D is a simple pole of ζ A ( · ; δ), and res( . If, additionally, D < N , the analogous statement and conclusion is true for the distance zeta function ζ A and we have ( Let us now introduce some additional definitions, which are adapted from [12] to the present, much more general, context of compact subsets of an arbitrary Euclidean space, R N (with N ≥ 1): The screen S is the graph of a bounded, real-valued, Lipschitz continuous function S(τ ), with the horizontal and vertical axes interchanged: S := {S(τ ) + iτ : τ ∈ R}. The Lipschitz constant is denoted by S Lip . Furthermore, we let sup S := sup τ ∈R S(τ ) ∈ R. For a compact subset A of R N , we always assume that the screen S lies to the left of the critical line {Re s = D}, i.e., that sup S ≤ D. Moreover, the window W is defined as W := {s ∈ C : Re s ≥ S(Im s)}. The set A is said to be admissible if its tube (or distance) zeta function can be meromorphically extended to an open connected neighborhood of some window W . [12,Def. 5.2]). An admissible compact subset A of R N is said to be relaxed d-languid if there exist λ > 0 and δ > 0 such that the function ζ λA (s) := λ s ζ A (s; δ) satisfies the following growth conditions: There exist real constants κ and C > 0 and a two-sided sequence (T n ) n∈Z of real numbers such that T −n < 0 < T n for n ≥ 1, lim n→∞ T n = +∞ and lim n→∞ T −n = −∞, satisfying the following two hypotheses, L1 and L2: L1 There exists c > dim B A such that |ζ λA (σ + iT n )| ≤ C(|T n | + 1) κ , for all n ∈ Z and all σ ∈ (S(T n ), c).
i.e., for every σ < c and, additionally, there exists a sequence of screens S m (τ ) :

Pointwise and distributional tube formulas and a criterion for Minkowski measurability
In this section, we state and sketch the proof of our main results, the pointwise and distributional tube formulas, valid for a large class of compact subsets of R N (see Thms. 2.1 and 2.2 below), along with an associated Minkowski measurability criterion (see Thm. 2.3). These results extend to higher dimensions the corresponding tube formulas and Minkowski measurability criterion obtained for fractal strings in [12], §8.1 and §8.3, respectively. We point out that the detailed proofs of our main results (stated in a much more general form and within the broader context of relative fractal drums) can be found in the long paper corresponding to this note, [11]. Moreover, we note that in light of (2), Thms. 2.1, 2.2 and 2.3 have an obvious analog for tube (instead of distance) zeta functions. Also, the exact tube formula stated in Thm. 2.1 has a counterpart with error term (much as in Thm. 2.2). Finally, we refer to [11] and [12, §13.1] for many additional references on tube formulas in various settings, including, [1-3, 5-8, 13, 16].
The key observation in deriving Thms. 2.1 and 2.2 below is the fact that the tube zeta function of a compact set A in R N is equal to the Mellin transform of its modified tube function f (t) := χ (0,δ) (t)t −N |A t |, where χ E denotes the characteristic function of the set E. More precisely, one has that ζ A (s; δ) = {Mf }(s) := ∞ 0 t s−1 f (t) dt, where M denotes the Mellin transform. One then applies the Mellin inversion theorem (see, e.g., [15,Thm. 28]) to deduce that |A t | = 1 2πi c+i∞ c−i∞ t N −s ζ A (s; δ) ds, for all t ∈ (0, δ), where c > dim B A is arbitrary. One then proceeds in a similar manner as in [12,Ch. 5] for the case of fractal strings. More precisely, one works with a k-th primitive function of t → |A t | in order to be able to represent the above 2. Clearly, P(ζA( · ; δ), U ) is a discrete subset of C and is independent of δ; hence, so is P(ζA( · ; δ), C). Therefore, we will often write P(ζA, U ) or P(ζA, C) instead. integral as a sum over the complex dimensions contained in the window W . Here, k ∈ N is taken large enough to ensure pointwise convergence of this sum. From this result, one then derives the distributional tube formula for every value of k (even for k ∈ Z), and, in particular, for k = 0. In this way, we obtain the fractal tube formulas expressed in terms of the tube zeta function and then use the functional equation (2) in order to translate them in terms of the distance zeta function.
Theorem 2.1 (Pointwise tube formula). Let A be a compact subset of R N that is relaxed strongly d-languid for some δ > 0, λ > 0 and κ < 1. Furthermore, assume also that dim B A < N . Then, for every t ∈ (0, min{δ, λ −1 , (λB) −1 }) the following exact pointwise tube formula is valid (where B is the constant appearing in L2' of Def. 1.3 above) : In the case when κ ∈ R, we usually only have a distributional tube formula. Furthermore, if A is only relaxed d-languid, we will have a distributional error term, with information about its asymptotic order given in the sense of [12, §5.4]. Namely, the distribution R ∈ D ′ (0, δ) is said to be of asymptotic order at most t α (resp., less than t α ) as t → 0 + if when applied to a test function ϕ ∈ D(0, δ), 3 we have that R, ϕ a = O(a α ) (resp., R, ϕ a = o(a α )), as a → 0 + , where ϕ a (t) := a −1 ϕ(t/a) (and the implicit constants may depend on ϕ). We then write that R(t) = O(t α ) (resp., R(t) = o(t α )) as t → 0 + . Theorem 2.2 (Distributional tube formula). Let A be a relaxed d-languid compact subset of R N , for some λ > 0, δ > 0 and κ ∈ R. Furthermore, assume that dim B A < N and denote by V(t) the distribution generated by t → |A t |. Then, we have the following distributional equality: More precisely, the action of V(t) on a test function ϕ ∈ D(0, δ) is given by If, in addition, A is relaxed strongly d-languid, then, for test functions in D 0, min{δ, λ −1 , (λB) −1 } , we have that R ≡ 0 and W = C; hence, we obtain an exact tube formula in that case.
One of the applications of the above results is a Minkowski measurability criterion for a compact relaxed d-languid subset of R N (see Thm. 2.3 below), which generalizes [12,Thm. 8.15] to higher dimensions. In the proof of Thm. 2.3, one direction is a consequence of the distributional tube formula (Thm. 2.2 above) and the uniqueness theorem for almost periodic distributions (see [14,§VI.9.6,p. 208]). The other direction follows from a generalization of the classic Wiener-Ikehara Tauberian theorem (see [4]). Theorem 2.3 (Minkowski measurability criterion). Let A be a compact subset of R N such that D := dim B A exists and D < N . Furthermore, assume that A is a relaxed d-languid set for a screen passing between the critical line {Re s = D} and all the complex dimensions of A with real part strictly less than D. Then, the following statements are equivalent: (a) A is Minkowski measurable.
(b) D is the only pole of the distance zeta function ζ A located on the critical line {Re s = D}, and it is simple.
We conclude this note by pointing out that, in a precise way, the above results generalize the corresponding ones obtained for fractal strings in [12, §8.1 & §8.3]. Namely, this can be seen from the fact that for the geometric zeta function ζ L of a nontrivial fractal string L = (l j ) j≥1 and the distance zeta function of the set A L := {a k := j≥k l j : k ≥ 1}, we have that ζ A L (s; δ) = s −1 2 1−s ζ L (s) + 2s −1 δ s , where δ > l 1 /2, and this identity holds on any subdomain U of C to which any of the two zeta functions has a meromorphic continuation; see [8, §2.1]. Hence, if U ⊆ C \ {0}, then ζ L and ζ A L have the same visible complex dimensions in U .