A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy

The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the structure of corresponding complex dimensions. Also, this tool is particularly effective in the setting of self-similar fractal subsets of $\mathbb{R}$ which have been shown to be Minkowski measurable if and only if they are nonlattice. This paper features a survey on the pertinent results of Lapidus and van Frankenhuijsen and a preliminary extension of the theory of complex dimensions to subsets of Euclidean space, with an emphasis on self-similar sets that satisfy various separation conditions. This extension is developed in the context of box-counting measurability, an analog of Minkowski measurability, which is shown to be characterized by complex dimensions under certain mild conditions.


1.
Introduction. Let A be a bounded set in Euclidean space R m . We address the box-counting content of A (see Definition 2.7), an analog of Minkowski content (see Definition 2.2) given by where D is the box-counting dimension of A and N B (A, ε −1 ) is the maximum number of disjoint closed balls with centers a ∈ A and radii ε > 0. If B(A) exists in (0, ∞), then A is said to be box-counting measurable; see Definition 2.7.

DETTMERS, GIZA, KNOX, MORALES, AND ROCK
Self-similar sets in R m provide the key setting for this paper. Let Φ = {ϕ j } N j=1 denote a self-similar system, an iterated function system in which each ϕ j is a contracting similarity acting on R m . Also, let F denote the unique nonempty compact set satisfying F = ∪ N j=1 ϕ j (F ), i.e., F is a self-similar set; see [7] as well as Definition 2.10. If the scaling ratio of ϕ j is denoted by r j and there are some positive real number r and positive integers k j such that r j = r kj , then Φ and F are said to be lattice. Otherwise, Φ and F are nonlattice; see Definition 2.26. Although the terminology is not used by Lalley in [9], the box-counting measurability of selfsimilar sets is studied therein. Also, some of the results in [9] are used in [20], the new results of which are presented for the first time in Section 4 of this paper.
A conjecture of Lapidus made in the early 1990s claims that, under appropriate conditions, a self-similar set is Minkowski measurable if and only if it is nonlattice. Such a conjecture was proven for subsets of the real line, under assumptions including that the self-similar system in question satisfies the open set condition (see Definition 2.16 as well as [3,7,21]) and the Minkowski dimension D of the attractor satisfies 0 < D < 1 (see the work of Falconer in [2] and Lapidus and van Frankenhuijsen in [17], as well as Theorem 3.25 in the present text). The conjecture of Lapidus asserts that the statement holds in R m for m ≥ 2 and self-similar sets of Minkowski dimension D where m − 1 < D < m. Gatzouras was able to prove that if a set F is a nonlattice self-similar set then F is Minkowski measurable; see [6]. The converse remains an open problem from which a substantial amount of active research has stemmed. For instance, see [8,12,13,19]. An analog of Gatzouras' result in terms of box-counting measurability for self-similar subsets of R m with m ≥ 1 is provided by Theorem 2.28 (a restatement of Theorem 1 in [9]). This theorem is also a key motivation for Corollary 5.5 below and other results of this paper.
Minkowski content has attracted attention due in part to its connection with the (modified) Weyl-Berry conjecture as proven in the context of subsets of R by Lapidus and Pomerance in [14]. This result establishes a relationship between the Minkowski content of the boundary of a bounded open set in R and the spectral asymptotics of the corresponding Laplacian. In turn this led to a reformulation of the Riemann hypothesis as an inverse spectral problem associated with bounded open subsets of R (see [11]) and the development of the theory of complex dimensions of fractal strings; see [17] as well as Section 3 below.
Also in [17], Lapidus and van Frankenhuijsen use complex dimensions to expand upon the lattice/nonlattice dichotomy of certain self-similar sets in R beyond Minkowski measurability. A fractal string is a nonincreasing sequence L = ( j ) ∞ j=1 of positive real numbers which tend to zero; see Definition 3.1. The set of complex dimensions of L, denoted by D L (W ), is the set of poles of a meromorphic extension of the Dirichlet series ∞ j=1 s j defined for suitable region W ⊆ C; see Definition 3.3. This Dirichlet series is called the geometric zeta function of L, and its meromorphic extension to any suitable region W is denoted by ζ L .
Given a bounded open subset of R denoted by Ω with infinitely many connected components, the lengths of these components constitute a fractal string. The complex dimensions of this fractal string are used, for instance, to characterize the Minkowski measurability of ∂Ω, the boundary of Ω; see Theorem 3.11 and [17]. In the case where ∂Ω is a self-similar set, the complex dimensions are given by the complex solutions of the corresponding Moran equation (as in (1)  where the scaling ratios of the self-similar system that define ∂Ω are given by the scaling vector r = (r j ) N j=1 . Note that, by Moran's Theorem (Theorem 2.22, see also [3,7]), the Minkowski dimension of ∂Ω in this case is the unique positive real number in S r . Moreover, deep connections between the structure of the complex dimensions of lattice and nonlattice self-similar sets in R, expanding the lattice/nonlattice dichotomy in this setting, are developed in [17,Chapter 3]. In particular, simultaneous Diophantine approximation is used therein to tremendous effect to explore the intricate structure of such complex dimensions.
In the present paper, we make use of many of the results of Lapidus and van Frankenhuijsen in [17] in terms of the complex dimensions associated with N B (A, ·), the box-counting function of a given bounded set A in R m (not just R). In particular, by making use of the box-counting fractal strings and developed by Lapidus, Zubrinić, and the fifth author in [16] (see Definition 4.1), we show that the boxcounting measurability of A is characterized by the structure of the corresponding complex dimensions. This result is presented in Corollary 5.5 as an analog of [17,Theorem 8.15] (which appears as Theorem 3.11 below). Additionally, we show that under mild conditions the box-counting content is given by where D is the box-counting dimension of A (see Definition 2.5), L is the boxcounting fractal string constructed using N B (A, ·), and res(ζ L (s); ω) is the residue of ζ L at ω ∈ W for a suitably defined W . Also, in terms of self-similar sets in R m that are either strongly separated or satisfy either the open set condition (see Definitions 2.19 or 2.16, respectively), we show that the corresponding complex dimensions are often given by elements S r ; see Corollaries 4.10 and 4.15. Various attempts to extend the theory of complex dimensions to sets in R m have been made. For instance, the approach taken in [12,13] involves similarly defined complex dimensions and the Minkowski measurability and tilings of self-similar sets that satisfy the open set condition along with certain a nontriviality condition, but this approach does not extend to other types of bounded sets in R m . In [15], a theory of complex dimensions is developed in the context of distance and tube zeta functions associated with arbitrary bounded sets in R m . However, as of the writing of this paper, the results presented therein have not been used to provide an extension of lattice/nonlattice dichotomy to self-similar sets in R m .
The structure of the paper is as follows. In Section 2, we summarize many of the classical results on Minkowski/box-counting dimension and self-similar sets, including a discussion of simultaneous Diophantine approximation and its connection to the lattice/nonlattice dichotomy. In Section 3, we summarize the results on fractal strings and complex dimensions of [17] that motivate and are used to prove the new results presented in later sections. In Sections 4, the new results of master's thesis [20] pertaining to box-counting fractal strings of self-similar sets in R m under various separation conditions are presented along with a couple of key examples. Section 5 concludes the paper with a discussion of a few results from the master's thesis [18], an application of which includes the criterion for box-counting measurability presented in Corollary 5.5. 2.1. Dimensions and contents. The following notation and terminology are wellknown in the literature on fractal geometry. Notation 2.1. Let A ⊆ R m and x ∈ R m . Let d(x, A) denote the distance between x and A given by d(x, A) := inf{||x − a|| m : a ∈ A}, where || · || m denotes the usual m-dimensional Euclidean norm. The notation d m denotes the m-dimensional Euclidean metric. For ε > 0, the open ε-neighborhood of A, denoted by A ε , is the set of points in R m within ε of A given by A ε := {x ∈ R m : d(x, A) < ε}. Also, m-dimensional Lebesgue measure is denoted by vol m .
When dim M A = dim M A, the corresponding limit exists and the common value, denoted by dim M A, is called the Minkowski dimension of A. In the case where D M = dim M A exists, the upper and lower Minkowski contents of A are respectively defined by If M * (A) = M * (A), the corresponding limit exists and the common value, denoted by M (A), is called the Minkowski content of A. If 0 < M * (A) = M * (A) < ∞, then A is said to be Minkowski measurable.
A well-known equivalent formulation of Minkowski dimension is box-counting dimension. As noted in [3] and elsewhere, the types of "boxes" used in the computation of box-counting dimension can vary. However, in this paper (as in [16,20]), only the specific notion of box-counting function defined below is considered.
denotes the maximum number of disjoint closed balls with centers a ∈ A and radii x −1 > 0.
Remark 2.4. Note that for ε > 0, N B (A, ε −1 ) denotes the maximum number of disjoint closed balls with centers a ∈ A and radii ε. Although it may seem unnatural to define the box-counting function in terms of x = ε −1 , this notation is used throughout [16] and [20] in order to make use of the results of [17], as done in Section 4 below. Hence, the convention is adopted here as well.
Definition 2.5. For a bounded set A ⊆ R m , the lower and upper box-counting dimensions of A, denoted by dim B A and dim B A, respectively, are given by When dim B A = dim B A, the corresponding limit exists and the common value, denoted by dim B A, is called the box-counting dimension of A.
The following theorem is a classic result in dimension theory. The terminology box-counting content does not seem to be used elsewhere in the literature. However, the concept (see Definition 2.7) is a key component of [9] and allows for content to be studied in the context of box-counting functions. Such an approach is central to [20], the new results of which are presented and expanded upon in Section 4 of this paper. Note that, at least intuitively, for a bounded set A ⊆ R m we have for some positive constant c that which is to say that the volume of the open ε-neighborhood of a set A is approximately equal to the product of the maximum number of disjoint closed balls of radius ε with centers in A and the "size" of each ball given by cε m .

2.2.
The Hausdorff metric and self-similar sets. Self-similar sets can be constructed as fixed points of self-similar systems, as described in the next two definitions. Let K denote the set of nonempty compact subsets of a Euclidean space R m equipped with d m . It is a well-known fact that K equipped with the Hausdorff metric is a complete metric space (see [4, 2.10.21]). Thus, Banach's Fixed-Point Theorem applies and establishes the existence of a unique attractor for Φ, see [5, §10.3].
Definition 2.10. A function ϕ : R m → R m is a contracting similarity on R m if for all x, y ∈ R m and some 0 < r < 1, called the scaling ratio of ϕ, we have d m (ϕ(x), ϕ(y)) = rd m (x, y).
A self-similar system on R m is a finite collection Φ = {ϕ j } N j=1 of N ≥ 2 contracting similarities on R m . By a mild abuse of notation, a self-similar system Φ may be interpreted as a map Φ : K → K defined by Φ(·) := N j=1 ϕ j (·). The scaling vector of Φ is given by r = (r j ) N j=1 where, for each j = 1, 2, . . . , N , the contracting similarity ϕ j has scaling ratio r j . The attractor of Φ is the unique nonempty compact set F ∈ K such that Φ(F ) = F . Also, a self-similar set is the attractor of a self-similar system.
Contracting similarities have the following characterization (see Proposition 2.3.1 of [7]) which is used in the proof of Theorems 2.33 and 4.12 below. Proposition 2.11. A function ϕ is a contracting similarity on R m with scaling ratio r if and only if ϕ(·) = rQ(·) + t, where Q is an orthonormal transformation, t is a fixed translation vector, and 0 < r < 1.
The following examples are revisited throughout the paper.
Example 2.12. The Cantor set C is the attractor of the self-similar system Φ C on R given by the following pair of contracting similarities: The scaling vector of Φ C is given by r C = (1/3, 1/3).
Example 2.13. Consider the self-similar system Φ φ on R given by where φ = (1 + √ 5)/2 (the Golden Ratio). The attractor of this self-similar system is a Cantor-like set denoted by A φ and its scaling vector is given by r φ = (1/2, 1/2 φ ).
Example 2.14. The Sierpiński gasket S G is the attractor of the self-similar system Φ S on R 2 given by the following three contracting similarities: The scaling vector of Φ S is r S = (1/2, 1/2, 1/2). See Figure 1.
be a self-similar system with attractor F . Then Φ and F are said to be strongly separated if the images of F under the contracting similarities ϕ j are pairwise disjoint. If, in addition, δ := sup{α : d(x, y) > α ∀x ∈ ϕ j (F ), y ∈ ϕ k (F ), j = k, j, k = 1, . . . , N } is positive and finite, then Φ and F are said to be δ-disjoint.
Remark 2.20. Note that strongly separated self-similar systems satisfy the (strong) open set condition and their attractors are always totally disconnected. Also, Φ C , Φ φ , and Φ 1 are strongly separated, but Φ S is not.
Remark 2.23. Given a scaling vector r = (r j ) N j=1 , we are also interested in the set of complex solutions of (1) denoted by In particular, these values provide the complex dimensions associated with many of the self-similar sets studied throughout this paper.
Example 2.24. By Theorem 2.22, the self-similar sets C, A φ , S G and F 1 in Examples 2.12, 2.13, 2.14, and 2.15 have the following box-counting dimensions: In particular, the box-counting dimension of A φ is given by the unique positive real number D φ = dim B A φ = log φ 2 that satisfies the following equation: where φ is the Golden Ratio. The approximation D φ ≈ .77921 follows from Lemma 3.29 of [17] and serves as an application of the simultaneous Diophantine approximation provided by Lemma 3.16 of [17]. See Section 2.5 below.
Remark 2.25. The subset of K comprising finite sets of vectors with rational components is countable and dense in K (which shows that K is separable). Hence, in the Hausdorff metric, any A ∈ K can be approximated as closely as one would like by a finite set. However, such an approximation does not correspond to a satisfactory notion of approximation for the dimensions of the corresponding sets. Indeed, finite sets are 0-dimensional, but a compact subset of R m can have Minkowski dimension anywhere in [0, m].
As discussed in Theorems 2.33 and 4.12, the approximation of a nonlattice selfsimilar set by a sequence of lattice self-similar sets in the Hausdorff metric, as well as the convergence of their box-counting complex dimensions in the sense described in Remark 3.27, follow from the simultaneous Diophantine approximation of the corresponding scaling vectors described in the next section.
2.4. Lattice/nonlattice dichotomy and measurability. Much of the remainder this paper focuses on the lattice/nonlattice dichotomy for self-similar sets in Euclidean space, an introduction to which is provided in this section. For the purposes of this paper, the dichotomy is studied in the context of (inner) Minkowski dimension on the real line in Section 3.2 and in the context of box-counting dimension in Sections 4 and 5).
Definition 2.26. Let Φ be a self-similar system with scaling vector r = (r j ) N j=1 and attractor F . Then Φ, its scaling vector r, and its attractor F are said to be lattice if there exist 0 < r < 1 and N positive integers k j such that gcd(k 1 , . . . , k N ) = 1 and r j = r kj for j = 1, 2, . . . , N . Otherwise Φ, r, and F are said to be nonlattice.
In the sequel, we write f (x) ∼ g(x) as x → a when lim x→a f (x)/g(x) = 1.
Remark 2.29. Part (a) of Theorem 2.28 says that nonlattice sets are box-counting measurable (see Definition 2.7) and part (b) says that lattice sets exhibit a logperiodic structure. Additionally, if for some lattice set F there exist β 1 , β 2 ∈ [0, h) such that C β1 = C β2 , then F is not box-counting measurable and F is said to exhibit geometric oscillations of order D = dim B F = D r . This is the case for the 1-dimensional set F 1 and the Sierpiński gasket S G with D = 1 and D = log 2

2.5.
Simultaneous Diophantine approximation. The simultaneous Diophantine approximation provided by Lemma 3.16 of [17] says that if at least one of a finite set of real numbers α 1 , . . . , α N is irrational, then these numbers can be approximated by rational numbers with a common denominator. Thus, one can find integers q such that for each j = 1, . . . , N , the product qα j is within a small distance to the nearest integer. This leads to Lemma 2.31. Note that this simultaneous Diophantine approximation also leads to the convergence of a sequence of lattice sets to a given nonlattice set in the Hausdorff metric (as in Theorem 2.33) as well as the convergence of corresponding complex dimensions (as in Theorem 4.12 below and described heuristically in Remark 3.27). Example 2.32. A simple yet illustrative example of the convergence described in Lemma 2.31 stems from the nonlattice set A φ in Examples 2.13 and 2.27 (see [17]). It is well known that φ is approximated by ratios of consecutive terms of the Fibonacci Figure 3, φ approximated by f M +1 /f M for M = 2, . . . , 9.
The sequence of lattice scaling vectors in the Lemma 2.31 give rise to a sequence of lattice sets which converge to a given nonlattice set.
j=1 be a nonlattice self-similar system on R m with attractor F and scaling vector r. Then there exists a sequence of lattice self-similar systems (Φ M ) ∞ M =1 with scaling vector r M and attractor F M for each M ∈ N such that each of the following statements holds as M → ∞: Proof. The key to the construction of the lattice sets F M lies in replacing the scaling ratios of a given nonlattice system Φ by those of the lattice scaling ratios stemming from an application of Lemma 2.31 as follows: (i) For each j = 1, · · · , N we have ϕ j (·) = r j Q j (·) + t j by Proposition 2.11.
(ii) For each j = 1, . . . , N and each M ∈ N, define r M,j using the simultaneous Diophantine approximation provided by Lemma 2.31.
for each j = 1, · · · , N . Also, define F M to be the attractor of Φ M . By Theorem 11.1 in Chapter III of [1], which (for our purposes) says that attractors of self-similar systems depend continuously on the scaling vectors, we have If Φ is strongly separated, then it readily follows from Definitions 2.9 and 2.19 that, for large enough M , Φ M is strongly separated. Also, assuming δ M does not tend to δ yields a contradiction in light of the fact that F M → F .

The Theory of Complex Dimensions of Fractal Strings.
3.1. Fractal strings, complex dimensions, and zeta functions. This section provides a summary of some of the results from the theory of fractal strings, complex dimensions, and zeta functions, with a focus on Minkowski measurability (in terms of inner Minkowski content and dimension) of certain subsets of the real line. See [17] for a full and thorough introduction to the theory of complex dimensions of fractal strings. These results both motivate and provide a foundation for the new results explored in Sections 4 and 5 below.
j=1 is a nonincreasing sequence of positive real numbers such that lim j→∞ j = 0. By a mild abuse of notation, one may think of a fractal string L as the multiset L = {l n : l n has multiplicity m n , n ∈ N}, where (l n ) n∈N is the strictly decreasing sequence of distinct lengths j = l n and the multiplicity m n is the number indeces j such that j = l n .
Remark 3.2. An ordinary fractal string Ω can be expressed as the countable union of pairwise disjoint open intervals. Throughout this work, as in [17], we assume an ordinary fractal string Ω comprises infinitely many connected components with lengths determining a fractal string L.
The dimension of a fractal string L, denoted by D L , is defined by D L = σ. The geometric zeta function of L, denoted by ζ L , is defined by where s ∈ C such that Re(s) > D L and the lengths l n and multiplicities m n are as in Definition 3.1. Note that ζ L is holomorphic on the half-plane Re(s) > D L (see [22,§VI.2]).
The following theorem is a restatement of Theorem 1.10 of [17] and justifies the notion of referring to D L as a dimension.   As such, the length of C 4 is given by Note that C 4 cannot be the attractor of a self-similar system. Additionally, we have dim M C 4 = 1 and C 4 is Minkowski measurable with Minkowski content M (C 4 ) = 1/2. However, as explained in Example 3.13, C 4 is not inner Minkowski measurable with respect to its inner Minkowski dimension given by dim iM C 4 = D L4 = 1/2. This fact is closely tied to the structure of the complex dimensions of L 4 , as indicated in Theorem 3.11.
The complex dimensions of a fractal string are defined as the poles of a meromorphic extension of its geometric zeta function. By a mild abuse of notation, both the geometric zeta function of L and its meromorphic extension to some window are denoted by ζ L .
One of the key results in the theory of complex dimensions for ordinary fractal strings is the criterion for Minkowski measurability provided by Theorem 3.11 below. This result involves the following counting function.   The following lemma is partially a restatement of Lemma 13.110 of [17], and a more general result is Proposition 5.2 below. A much stronger result, which is beyond the scope of this paper, holds in the setting of Mellin transforms and generalized fractal strings (viewed as measures) as described in [17]. In [17, §5.3], the terms languid and strongly languid describe the growth of a geometric zeta function ζ L in terms of three technical conditions called L1, L2, and L2 . L1 is a polynomial growth condition along horizontal lines (in the complex plane) necessarily avoiding the poles of ζ L , L2 is a polynomial growth condition along the vertical direction of a corresponding screen, and L2 is a stronger version of L2. A fractal string L is languid if ζ L satisfies L1 and L2, and is strongly languid if ζ L satisfies L1 and L2 . These conditions allow for some of the key results in [17] to hold, such as Theorem 8.15 therein, which appears here as Theorem 3.11. This theorem is a primary motivation behind the new results presented in Section 4 and especially Corollary 5.5 and Proposition 5.7. (See also [15,16,18,20].) Example 3.13. By Example 3.6, we have for Re(s) > D L4 = 1/2. The closed form on the right-hand side of this equation allows for a meromorphic extension of ζ L4 to all of C and it is used to verify that L 4 is strongly languid. It follows that Note that D L4 = 1/2 is not the only complex dimension with real part equal to 1/2, so by Theorem 3.11, the fat Cantor set C 4 is not inner Minkowski measurable (even though it is Minkowski measurable, see Example 3.6).
Example 3.14. The Cantor string is the ordinary fractal string given by Ω CS := [0, 1]\C where C is the Cantor set. The corresponding fractal string L CS (also referred to as the Cantor string) is given by L CS = {1/3 n : 1/3 n has multiplicity 2 n−1 , n ∈ N}.
The geometric zeta function of the Cantor string, denoted by ζ CS , is given by for Re(s) > D L C S = dim M C = log 3 2. The closed form on the right-hand side of this equation allows for a meromorphic extension of ζ CS to all of C and it is used to show that L CS is strongly languid. It follows that the set of complex dimensions of the Cantor string is given by We have that D CS := D L CS = log 3 2 = dim B C. Moreover, D CS is not the only complex dimension with real part equal to D CS , so by Theorem 3.11, the Cantor set C is not Minkowski measurable. This fact was established in [14] via the equivalence of (ii) and (iii) and showing that (ii) does not hold. Actually, in [14], M * and M * are explicitly computed and shown to be different (with 0 < M * < M * < ∞).
Note that in part (i) of Theorem 3.11, the only complex dimensions of interest are the ones which have real part equal to D L . This motivates the following definition, which agrees in spirit with the definition of principal complex dimensions of [15].  For an ordinary fractal string Ω with lengths L, the complex dimensions of L provide more than just a criterion for the Minkowski measurability of the boundary ∂Ω. In particular, if L is strongly languid, the geometric counting function N L can be written as a sum over the complex dimensions of the residues of ζ L as in the following theorem, a special case of Theorem 5.14 in [17].
Theorem 3.17. Let L be a strongly languid fractal string. Then, for some a > 0 and all x > a, the pointwise explicit formula for N L is given by where the term in braces is included only if 0 ∈ W \D L (W ). If, in addition, all of the principal complex dimensions of L are simple, then for x > a we have where D = D CS = log 3 2, p = 2π/ log 3, and n = [log 3 x] where [y] denotes the integer part of y ∈ R. Note that in [17], this formula is derived directly using a particular Fourier series. Nonetheless, the formula for N CS provided in (5) also follows from an application of Theorem 3.17 since each complex dimension ω = D + ikp ∈ D CS is simple and the residue of ζ CS at each ω is independent of k ∈ Z. The common value of these residues is given by res(ζ CS (s); D + ikp) = 1 2 log 3 . Figure 2. The (nearly) log-periodic structure of N CS (x)/x D can be seen in this figure, which is indicated by the geometric oscillations of order D inherent to Ω CS . Also, the fact that the Cantor set C is not Minkowski measurable (as discussed above in Example 3.14) can be inferred from this figure 3.2. Lattice/nonlattice dichotomy of self-similar strings. Many results regarding the special case of the lattice/nonlattice dichotomy for (nontrivial) selfsimilar subsets of the real line have been established. See [17].  j=1 r j < 1, then the nonempty ordinary fractal string Ω = I\F is called a self-similar string. If Φ is lattice (or nonlattice), then Ω is a lattice (or nonlattice) string.

Remark 3.20.
Let Ω = I\F be a self-similar string. Let K denote the positive number of connected components in I\(∪ j ϕ j (I)) which have positive length, and let g k L denote the length of the kth connected component for k = 1, . . . , K arranged so that 0 < g 1 ≤ · · · ≤ g K < 1 and  Let Ω be a self-similar string (as in Definition 3.19) with lengths L. Then the geometric zeta function ζ L has a meromorphic extension to the whole complex plane given by Here, ζ L (1) = L is the total length of Ω as well as the length of the interval I.  (1) given by (2). Also, each complex dimension has a multiplicity at most that of the corresponding solution. If, in addition, Ω has a single value for the gaps g 1 = · · · = g K , then D L = S r .
Example 3.23. The Cantor string Ω CS and the Golden string Ω φ := [0, 1]\A φ are self-similar strings, each with a single gap. So, Corollary 3.22 applies to Ω CS and Ω φ , and the complex dimensions are given by S r C and S r φ , accordingly. The set of complex dimensions D CS of the Cantor string Ω CS (or L CS ) is determined in Example 3.14. The complex dimensions D L φ of the Golden string Ω φ are the solutions of the transcendental equation See Figure 3 for images of successive approximations of the complex dimensions of the Golden string. These images were not obtained through solving (6) directly but rather through the approximation of the complex dimensions, stated in terms of the structure of roots of Dirichlet polynomials, as detailed in Chapter 3 of [17] and described heuristically in Remark 3.27. Note that Ω 4 is not a self-similar string since C 4 is not a self-similar set. Nonetheless, the lengths L 4 are the lengths of some self-similar string.
The complex dimensions of self-similar strings and the box-counting complex dimensions of many self-similar subsets of some Euclidean space (see Section 4) are often given by the set of complex solutions of Moran equations of the form (1). These sets are denoted by S r and defined in (2).
The following theorem is a small part of Theorems 3.6 and 3.23 in [17] which provides a wealth of information regarding the structure of the set S r . where z = e −ω log r −1 , m u is the number of j such that r j = r ku , and M is the number of distinct values among the r j . Hence there exist finitely many solutions ω 1 , . . . , ω q such that where p = 2π/ log r −1 .
If r is nonlattice, then ω = D r is the only element of S r with real part equal to D r and all others have real part less than D r . Also, there exists a sequence of elements of S r approaching Re(s) = D r from the left.
The Minkowski measurability of the boundary of a self-similar string is directly related to whether the string is lattice or nonlattice. The following theorem is a combination of the results stated in Theorem 8.23 and 8.36 from [17]. Example 3.26. The Golden string is nonlattice, and as such Theorem 3.25 implies that A φ in Minkowski measurable. Thus, Theorem 3.24 implies that set of principal complex dimensions of the Golden string is a singleton comprising D L φ ≈ .77921. That is, dim P C L φ = {D L φ }. Moreover, D φ is a simple pole of ζ L φ , so Theorem 3.11 applies and the (inner) Minkowski content of A φ is given by (3).
Remark 3.27. Chapter 3 of [17] provides a thorough description of the manner in which the set of roots of a nonlattice Dirichlet polynomial are approximated by the set of roots of a lattice Dirichlet polynomial. In this paper, this approximation is discussed in terms of sequences with convergence denoted by S r M → S r as M → ∞ and loosely described as follows: Given a nonlattice scaling vector r and any fixed T > 0, there is a lattice scaling vector r M (constructed through Lemma 2.31) such that each of the roots in S r with imaginary part less than T (in absolute value) is approximated in a uniform manner by a root in S r M and the multiplicity of the corresponding roots coincide. Moreover, the oscillatory period p (i.e., the period in the imaginary direction) of the roots in S r M with maximal real part D is much smaller than T . See Figure 3 for a collection of images which show the approximation of the roots in S r φ associated with the nonlattice scaling vector r φ . By Corollary 3.22, we have S r φ = D L φ .
The approximations seen in Figure 3 are given by a sequence of roots stemming from lattice scaling vectors whose components depend directly on the ratios of Fibonacci numbers, see Example 2.32. Note that, by Corollary 3.22, the convergence S r M → S r as M → ∞ described above also describes the "quasiperiodic" behavior of the structure of the complex dimensions of a self-similar string. As noted above, the approximation used in the proof of Theorem 2.33 makes explicit use of Lemma 2.31 which, in turn, brings along the convergence of complex dimensions in the context of self-similar strings as described in Remark 3.27 and Figure 3. This begs the question as to whether the convergence of complex dimensions also holds in suitable context for self-similar subsets of any Euclidean space and not just subsets of R (as is the case for self-similar strings). A potentially suitable context is described in the remainder of the paper by making use of the results presented in this section. In particular, one of the goals of studying box-counting zeta functions in Sections 4 and 5 is to find a framework in which the lattice/nonlattice dichotomy can be discussed in terms box-counting complex dimensions.

4.
Box-Counting Zeta Functions of Self-Similar Sets.

4.1.
Box-counting fractal strings and zeta functions. The material presented in this section follows from the results of Lalley in [9] as discussed in Sections 2.4 and 2.5, along with those determined by the third author in [20]. The work done in [20] was motivated by that of Lapidus and van Frankenhuisjen in [17] (as outlined in Section 3) and the box-counting fractal strings and zeta functions introduced by Lapidus,Žubrinić, and the fifth author and in [16].
That is, l −1 n is the positive real number where N B jumps from M n to M n+1 . Now, let m 1 = M 2 , and let m n = M n+1 − M n , for n ≥ 2. The box-counting fractal string of A, denoted by L B , is the fractal string with distinct lengths (l n ) n∈N and corresponding multiplicities (m n ) n∈N .
The following technical proposition and lemma were introduced in [16] as part of the development of a theory of box-counting fractal strings, zeta functions, and complex dimensions. In particular, they allow one to make use of the results of Lapidus and van Frankenhuijsen in [17] (as outlined in Section 3) in the context of box-counting functions, dimension, and content.  of N B (A, ·), and for The following is one of the key results of [16] and, in part, motivates the definition of box-counting zeta function given just below.   where Re(s) > 1 and ζ(s) denotes the Riemann zeta function. It is well-known that ζ converges for Re(s) > 1 and has a simple pole at s = 1, so D L B = dim B [0, 1] = 1.

4.2.
Self-similar sets under separation conditions. This section focuses on some the results of [20] which, in part, follow from results of [9]. All of the results presented here make use of either the (strong) open set condition or strongly separated self-similar systems.
Lemma 4.7. Let Φ be a δ-disjoint self-similar system with attractor F and scaling vector r = (r j ) N j=1 . If x > δ −1 , then N B (F, x) = N j=1 N B (F, r j x). Moreover, for any x > 0, where L(x) is a nonpositive, nondecreasing integer valued step function with a finite number of steps that is bounded below by 1 − N and vanishes for x > δ −1 .
Proof. A ball with radius x −1 ≤ δ and center in ϕ j (F ) does not intersect any other ϕ k (F ) for k = j. This fact, combined with the self-similarity of F , imply that The properties of L(x) follow readily from Lemma 4.2.
The following theorem provides a (nearly) closed form for the box-counting zeta function of a self-similar set whose counting function satisfies a renewal equation of the form (7).
for an at most countable number of values x ≥ x 1 and N L B (x) = 0 for x < x 1 , applying Lemma 3.10 along with (7) yields Let E(s) = s Since 0 < r j < 1 implies r j x 1 < x 1 and we have N B (F, u) = 1 for u < x 1 . Also, By combining (9), (10), and (11) we get Solving for ζ B (s) and simplifying yields (8). Hence, the Principal of Analytic Continuation allows ζ B to extend so as to be holomorphic on the half-plane Re(s) > D r = dim B F (see [22,§VI.2]).
The following corollary is a key step in the development of a lattice/nonlattice dichotomy from the perspective of the theory of (box-counting) complex dimensions associated with strongly separated self-similar systems. Corollary 4.9. Let Φ be a δ-disjoint self-similar system with attractor F and scaling vector r = (r j ) N j=1 , and let L B be the box-counting fractal string of F . Then the box-counting zeta function of F has a closed form given by where Re(s) > D r = dim B F and the values of m, n, the e k and the y k satisfy the following properties: m, n ∈ N with m ≤ n; with 1 − N = e 1 < · · · < e n < 0 with e k ∈ Z; 0 < y 1 < · · · < y n ≤ δ −1 ; and m ∈ N is the smallest number such that y m > x 1 .
The proof is omitted but can be found in [20]. Essentially, it follows from a careful decomposition of renewal equation of (7) and the evaluation of the integral defining E(s) in Theorem 4.8. The values of e k and y k are determined by the structure of L(x), which is a step function for 0 < x ≤ δ −1 , as provided by Lemma 4.7.
Corollary 4.10 provides a couple of results regarding the complex dimensions of a strongly separated self-similar set. In this corollary and Theorem 4.12, the numerator on the right-hand side of (12) plays a role in determining the structure of these complex dimensions of strongly separated self-similar sets. So, given a δ-disjoint self-similar set F , let h denote the numerator of (12) given by where s ∈ C has large enough real part.  Under certain conditions, the box-counting complex dimensions of a strongly separated nonlattice set are approximated by those of lattice sets.   Proof. By Proposition 4.13, we have N B (F, x) = N j=1 N B (F, r j x) + L(x) where |L(x)| ≤ γx D−ε for some constants γ, ε > 0. Now, temporarily let s denote a real number such that s > D − ε. Since D − ε − s < 0, we have −γs D−ε−s > 0. Therefore, since lim a→∞ a D−ε−s = 0, we have Thus, by allowing s ∈ C we have that E(s) converges for Re(s) > D − ε.
Under the hypotheses of Proposition 4.14 and Theorem 4.8, the principal complex dimensions are solutions of the corresponding Moran equation.  Example 4.16. The lattice set F 1 from Example 2.15 stems from the δ-disjoint selfsimilar system Φ 1 (with δ = 1/2)). The box-counting function of F 1 is determined in [16]. For 0 < x ≤ 2, we have Hence, M 1 = 1, M 2 = 2, and M 3 = 3. See Figure 1. For x > 2 and n ∈ N we have It follows that F 1 is not box-counting measurable. We have that dim B F 1 = 1 by Moran's Theorem (Theorem 2.22). So, considering that the extreme behavior of N B (F 1 , x)/x occurs at the endpoints of the intervals defined in (14), we also have x 1 = lim n→∞ 4 n 2 · 4 n−1 = 2, and Hence, F 1 exhibits geometric oscillations of order dim B F 1 = 1.
Since Φ 1 is δ-disjoint with δ = 1/2, Lemma 4.7 implies that the box-counting function of F 1 satisfies the renewal equation So, by Corollary 4.9 (and in agreement with the results of [16]), It follows that the box-counting complex dimensions of F 1 (and the principal complex dimensions) are given by Additionally, each of these complex dimensions is a simple pole, so Theorem 3.17 applies and a pointwise explicit formula for N B (F 1 , x) is given by (4).
Example 4.17. The Sierpiński gasket S G is the attractor of the lattice self-similar system Φ S . This system satisfies the open set condition but is not strongly separated (see Examples 2.14 and 2.18). The box-counting function of S G is given by As in the case of F 1 , the Sierpiński gasket S G is not box-counting measurable. We have that dim B S G = log 2 3 =: D. So, considering that the extreme behavior of N B (S G , x)/x D occurs at the endpoints of the intervals defined in (15) Hence, S G exhibits geometric oscillations of order dim B S G = log 2 3. Note that the box-counting function of S G satisfies the renewal equation Observe that for 0 < x ≤ 2, L(x) = −2, and for 2 < x ≤ 4, L(x) = 0. Now suppose that 2 n < x ≤ 2 n+1 for n ≥ 2. Then N (S G , x) = (3 n +3)/2. Since 2 n−1 < x/2 ≤ Additionally, each of the principal complex dimensions is a simple pole, so Theorem 3.17 applies and a pointwise explicit formula for N B (S G , x) is given by (4).

5.
Related Results and Future Work.

5.1.
Generalized content and zeta function. This section provides a preliminary framework for the study of complex dimensions in the setting of a simple class of Dirichlet type integrals (see [15] and references therein) as developed in [18]. Note that in the setting developed here there is no underlying geometry.
Additionally, the zeta function of f , denoted by ζ f , is given by If ζ f has a meromorphic extension to all of C, then D f := D f (C) denotes the set of complex dimensions of f .
The following theorem is essentially an analog of Lemma 13.110 of [17] (Lemma 3.10 in this paper), Theorem 13.111 of [17], and Lemma 3.13 of [16]. x D f . If C * = C * , then the content of f , denoted by C , is defined to be this common value. Additionally, if 0 < C * = C * < ∞, then f is said to be steady.
The following theorem is a type of generalization of Theorem 3.11 stated in terms of steady functions, but without any geometric context such as Minkowski or boxcounting measurability. The proof is omitted since it follows that of Theorem 8.15 in [17], mutatis mutandis. Theorem 5.4. Let f be such that ζ f is languid for a screen passing between the vertical line Re(s) = D f and all the complex dimensions of f with real part strictly less than D f and not passing through zero. Then the following are equivalent: (i) D f is the only complex dimension with real part D f , and it is simple.
(ii) f (x) = E · x D f + o(x D f ) for some positive constant E.
(iii) f is steady.
Theorems 3.17 and 5.4 combine to yield the following corollary. Proposititon 5.7 closes the paper and adds to the lattice/nonlattice dichotomy of self-similar sets from the perspective of box-counting measurability.
Proposition 5.7. Suppose Φ is a nonlattice strongly separated self-similar system with scaling vector r, attractor F , and dim B F = D r . Further, suppose h(s) = 0 if and only if s / ∈ S r . Then F is box-counting measurable and where h(s) is given by (13).
Proof. By Theorem 2.28(a), F is box-counting measurable since Φ is nonlattice. Also, Theorem 3.9 applies and we have that D r is a simple pole of ζ B . Since D r ∈ S r , we have h(D r ) = 0 and the residue of ζ B at D r is readily given by res(ζ B ; D r ) = h(D r ) N j=1 r Dr j log r −1 j .