Zeta functions and solutions of Falconer type problems for self-similar subsets of Z n.

Zeta functions and solutions of Falconer type problems for self-similar subsets of Z n. Driss Essouabri and Ben Lichtin Abstract. This paper uses the method of zeta functions to solve Falconer type problems about sets of k simplices whose endpoints belong simultaneously to a self similar subset F of Z n (k n) and a disc B(x) of a large radius x. Assuming that the similarity transformations pairwise commute, we study four Euclidean metric invariants of the simplices, the most basic (and frequently studied) of which is the distance between endpoints of a 1 simplex. For each, we introduce a zeta function, derive its functional properties, and apply such information to derive a lower bound on the upper Minkowski dimension of F which guarantees that the number of distinct metric invariants must be unbounded as x. Mathematics Subect Classifications: 11M41, 28A80, 52C10, 11J69, 11M32 Key words: self-similar discrete sets, fractals, zeta functions, meromorphic continuation, Minkowski dimension, Erdös distance problem, n simplices in fractals. 1 Introduction This article uses zeta function methods to study the discrete analog of Falconer type problems for unbounded self similar sets F Z n. Given F, a Euclidean metric invariant defined on the set of k simplices (1 k n), and a disc B(x) of radius x, a (discrete) Falconer type problem asks for a threshold on the upper Minkowski dimension of F, above which the number of distinct invariant values, determined by simplices whose endpoints belong to F B(x), must be an unbounded function of x when x. As such, a solution, for our purposes, is always asymptotic in nature. Our basic observation is that zeta function methods are very well suited to solve such problems when F is compatible, that is, a self similar set whose similarities pairwise commute. For each of the four invariants defined below, we prove that the number of distinct invariants must be, at least, an explicit positive power of # ( F B(x) ) if the upper Minkowski dimension of F exceeds a simple lower bound that depends only upon n. We are able to prove our theorems by combining the analysis of an underlying zeta function with diophantine methods over Z that estimate the density of integral points on classes of fairly simple algebraic sets. Indeed, to find each of the lower bounds (for the upper Minkowski dimension) needed to solve the Falconer problem, it suffices to impose the condition that a certain difference between a quantity analytic in nature and a quantity diophantine in nature must be positive. PRES Université de Lyon, Université Jean-Monnet (Saint-Etienne), Institut Camille Jordan (UMR 5208 du CNRS), Faculté des Sciences et Techniques, Département de Mathématiques, 23 rue du Docteur Paul Michelon, 42023 Saint-Etienne Cedex 2, France. Email: driss.essouabri@univ-st-etienne.fr Email: lichtin@frontier.com 1

We first choose a vector m = (m 1,..., m k+1 ) F k+1 ( m i 0 i) and define the k simplex Σ k (m) = m 1 m k+1,..., m k m k+1 := { η l (m l m k+1 ) : η l = 1 and η l 0 l }. Four different metric invariants of Σ k (m) interest us: l k l 1. (k = 1) Distance between two points. 2. (k = 2) The angle of a 2 simplex determined by the two unit vectors m 1 m 1 and m2 m. 2 3. (k 2) The configuration rooted at m k+1 for any k 2, that is, the vector of individual lengths (from m k+1 ) ( m 1 m k+1,..., m k m k+1 ), where each distance is assumed to be positive. 4. (k = n) The volume of Σ n (m). We have chosen these four invariants because they offer a good illustration of how zeta functions can be used to address Falconer type problems, a point that does not yet seem to have been observed in the literature. Other invariants can also be analyzed via this technique. Because these invariants have also been studied by other techniques, their study also allows us to compare the relative utility of our method. The statements of all the results we prove in the article are given in 2.1. It is important to realize that all our results are unconditional and apply for any n 2, except for that involving angles which requires n to be at least 4. This is in distinct contrast to the conditional results we described in [EL-1] and [EL-2], which assumed that results of Mattila could extend to unbounded discrete self similar sets. By restricting our attention to subsets of Z n, and using diophantine arguments that estimate the number of integral points on some simple algebraic sets of small degree we avoid having to impose any supplementary hypothesis from geometric measure theory. An overview of the results follows a description of the context in which we think it is helpful to place them. The original context for Falconer type problems is the geometric measure theory of bounded sets E R n with positive Hausdorff measure. The inspiration of a large body of work has been to prove a well known conecture of Falconer which predicts that a certain lower bound on the Hausdorff dimension of E guarantees that the distance set has positive Lebesgue measure. (E) = { x 1 x 2 : x 1, x 2 E} For dimensions n > 2, there is, however, no particularly compelling reason to be complacent and limit oneself to lengths of line segments. In particular, it would seemingly be more natural to work with simplices and replace distance by volume or configuration sets. The articles [GI], [GIM], [GGIP]), [IMP] have all been motivated to do ust this by formulating and proving variants of Falconer type problems, for example, for V ol(e) = {vol(σ n (x)) : x = (x 1,..., x n ) E n }. The question then becomes how large must the Hausdorff dimension of E be to insure that the Lebesgue measure of V ol(e) is positive. One other goal of such work has been to answer the same discrete Falconer type problem that is addressed in this paper, albeit for different classes of unbounded discrete sets E (neither self similar 2

nor contained in Z n ). In order to apply any result about bounded sets to discrete unbounded sets E, the idea common to all these papers is to reduce the problem to families of sufficiently small neighborhoods of bounded discrete sets by a natural scaling/normalization procedure applied to a subset E k of cardinality k. This offers the opportunity to apply a large arsenal of powerful tools from both harmonic analysis and geometric measure theory to the uniformly bounded neighborhoods. One can then try and bootstrap one s way back to say something about the number of distinct metric invariants determined by points in E k as k. This procedure has been useful, but in order for it to work one also needs to impose an additional hypothesis of a geometric measure theoretic nature. In particular, for each of the applications to discrete sets in the papers cited above it is necessary to impose the condition called s adaptability (for s larger than the Hausdorff dimension of the neighborhoods). Such a property, while plausible or technically appealing, does not seem all that easy to verify starting with unbounded self similar sets of the type we study. One consequence of our work is that it seems well suited to address discrete analogues of Falconer type problems without having to impose a priori the s adaptibility hypothesis. For our purposes here, it suffices for the set to be compatible, a property very easy to verify, in order to exploit the zeta function method. Furthermore, we believe that comparable results can also be proved for self similar subsets of Z n that are not compatible since we do not yet see a fundamental obstacle to extending the methods in [EL-1] to non compatible self similar sets. Indeed, even for such a set, there is a meromorphic continuation of its zeta function, as well as a generalized Dirichlet polynomial (see 2.2.1) whose roots include the poles of the zeta function belonging to some neighborhood of the boundary of absolute convergence. What still remains to be done is to prove Lemma 1 ( 2.2.1) for this set of roots (see the Remark at the end of 2.2.2 for some additional details). We hope to be able to say more about this in the near future. Our first result addresses the distinct distance problem of Erdös in an asymptotic sense and for any n 2. This seems to us a good first test case to see what the zeta function method is capable of. When F Z 2 is compatible, the exponent we derive for the lower bound of the number of distinct distances between points in F B(x) for all large x is almost as good as that predicted by Erdös, that is, 1 ε for any ε > 0. However, we do not see any way by which our method could do any better than this and give the actual exponent appearing in Erdös original paper. It is perhaps also worthwhile to point out here, since there may be some confusion on this point, that a recent article of Orponen [O] shows that the Hausdorff measure of the distance set determined by a self similar subset of R 2 equals 1. However, this paper does not show that the Lebesgue measure of the distance set is positive, a property that is needed to exploit the conversion method of Iosevich- Laba [IL], which shows how a solution of the Falconer conecture implies a solution to the Erdös distance problem (in the stronger non asymptotic sense). As a result, the result of Orponen does not yet seem to prove the Erdös distinct distance conecture for self similar subsets in the plane. When n > 2, we show that the zeta function method is capable of improving upon the (implicit) asymptotic result [SV] when the upper Minkowski dimension of F belongs to the interval (n n2 4 n 2 1, n]. On the other hand, if the upper Minkowski dimension is strictly less than n n2 4 n 2 1 then our exponent of growth is weaker than that in [ibid.]. These statements are given in Theorem 1 and proved in 3.1. It is important to emphasize that this result is only asymptotic in nature. It has nothing to say about the much more difficult question asked by Erdös about finite sets of any cardinality. The celebrated result proved by Guth-Katz [GK] evidently implies an asymptotic result, and in far greater generality. The same can be said about our result for n > 2 insofar as the work of [SV] is concerned. That 3

article applies for any finite set, so it clearly implies our asymptotic result for almost all values of the upper Minkowski dimension. What is of most interest in Theorem 1, is that our result finds a small interval for the upper dimension in which the exponent we derive for the asymptotic problem is actually superior to that given in [ibid.]. A natural variant of considering the length of a 1 simplex determined by m 1, m 2 F B(x) is the angle of the 2 simplex formed by connecting the origin to the two unit vectors in the directions of m 1, m 2. Assuming n 4, we derive a simple lower bound on the upper Minkowski dimension which insures that the number of distinct cosine values must grow, at least, to some positive power of #F B(x). As was noted in [IMP], little is known, in general, about the distribution of angles determined by points in unbounded discrete subsets of R n once n 4. Whereas the results proved in [ibid.] require the hypothesis of s adaptability, we give a solution to a Falconer problem for angles for any compatible self similar subset of Z n once n 4 with no other hypothesis. The statement is given in Theorem 2 and proved in 3.2. It is perhaps of interest to note here that the diophantine component of our proof only uses an elementary result about lattice points on absolutely irreducible quadratic hypersurfaces. The metric invariants in (3), (4) are examples of how one might hope to extend the Erdös distance problem to study the metric geometry of configurations or simplices determined by more than two points in higher dimensional contexts. Theorems 3 and Corollary 1 present our results concerning distinct numbers of configurations. Proofs are in Section 3.3. Here the technique is similar to that used in 3.1. Sections 4 and 5 study the distinct volume problem (4). The statement of the result is given in Theorem 4 and the proof is given in Section 5. The point of departure to prove all our results is based upon an extension of a classic Tauberian procedure for generating functions in the form of one variable Dirichlet series with positive coefficients f(s) = n a n λ s n ( where 0 < λ 1 < λ 2 <... is an unbounded sequence) to Dirichlet series in two or more variables whose coefficients are the values of the invariants (1)-(4) that interest us. For example, for invariant (1) this leads to a two variable series ζ 2 (s 1, s 2 ) := m 1 m 2 2. (1) m 1 s1 s2 m 2 (m 1,m 2) (F {0}) 2 In the classic case, the weakest effective description of a dominant asymptotic behavior (for all large x) of the average is given by A(x) := a n = Cx α ln l x + o(x α ln l x) where C 0. (2) λ n<x This follows once one knows that f(s) is absolutely convergent in a halfplane, has a meromorphic continuation to some larger region and has a single real pole on the boundary of analyticity. The value of α equals this pole. The idea is to say something equally precise about both the meromorphic behavior of the several variable series outside a domain of absolute convergence and the asymptotic behavior of the averages of their coefficients. The first requires us to identify precisely a certain finite set of extremal points on the boundary of the domain of analyticity. The set of such points is the natural analogue of the largest real pole for a one variable Dirichlet series. This is because they all belong to the vertex set of a polygon (or more 4

generally a polyhedron) that equals the real part of the boundary of analyticity of the pertinent Dirichlet series. We then refer to such points as belonging to the initial polar locus of the series. The second part is Tauberian, and aims to extend (2) in a natural way in order to describe a precise asymptotic lower bound for the averages of the coefficients of the several variable Dirichlet series. A lower bound suffices since the solution to a Falconer type problem only requires a nontrivial asymptotic lower bound for the number of distinct invariant values. The multivariate Tauberian theorem we prove in Theorem 8 (see 2.3) is both new and essential to our method. We are unable to find anything else in the literature that appears even remotely capable of proving the fundamental lower bound estimate (47), which is used to prove all six results stated in 2.1. An intriguing feature of this approach, it seems to us, is the fact that this classical based method, when combined with the diophantine geometric upper estimates for lattice points on spheres, can almost prove, in an asymptotic sense, the lower bound predicted by the Erdös conecture for compatible self similar sets inside Z 2. This is due to the fact that the lower bound on the upper Minkowski dimension of F only needs to be larger than n 2 for our result to follow. When combined with our, admittedly small, improvement in [SV], this suggests that the method can be helpful to address similar problems for sets in Euclidean spaces of any dimension, and for metric invariants (as well as other Riemannian metrics) that are definable by other polynomial (or analytic) expressions. Our work on angles, configurations, and volumes is evidence of its potential applicability. To appreciate further the content here, it is useful to emphasize an important difference, from our perspective, between the problems involving distance and volume. The source of the difference is that the appropriate zeta function to use for volumes, which we call a simplex zeta function (see 4), has an initial polar locus that is rather more difficult, in general, to describe precisely. This is critical because without a sufficiently precise description, it is usually not possible to use zeta function methods to prove explicit (and nontrivial) lower bound asymptotic estimates of the type needed to solve a Falconer type problem. This obstacle obliges us to introduce a property, which we call thickness. A self similar set that is thick does have a simplex zeta function for which the needed precision (in the initial polar locus description) is available. This is the point of the discussion in 4.2 (see Theorem 9). What is not, however, evident is how common a property is thickness, or if it even occurs at all. The purpose of 4.3 is to exhibit three very concrete examples of discrete self similar sets that are thick. It merits remarking here that any Pascal triangle mod p is a thick compatible self similar set. This property, curiously enough, does not seem to have been thought of before in the literature devoted to this much studied fractal like obect. Theorem 5 gives an asymptotic result about this family of fractal like sets that is new. Our final result, Theorem 6 extends this result to all Pascal pyramids mod p where multinomial coefficients mod p replace binomial coefficients, and where p is sufficiently large relative to n. Section 5 then applies the preceding discussion to the metric invariant of volumes in (4). The question, to which we provide at least one reasonably general answer, is the following discrete Falconer type problem: Given a discrete self similar set F, how small can the upper Minkowski dimension of F be so that the number of distinct volumes of n simplices, whose endpoints belong to F B(x), increases without bound as x. When F Z n is both thick and compatible, we answer this question in 5 (Theorem 4). It suffices for the upper Minkowski dimension to be larger than n 1. This is, in a general sense, the most complete result possible because any n simplex of a self similar set that lies inside a hyperplane has 5

exactly the same volume, namely 0. Such a self similar set can only have upper Minkowski dimension at most n 1. This result can be thought of as a discrete version of Theorem 1.6 [GIM] that is proved for bounded Salem subsets of R n. One possible way to think of this comparison is that the Salem property does for Fourier based methods what thickness does for zeta function methods. Unlike the arguments of 3, where the diophantine component of our method relies upon uniform, and very good, upper bound estimates for the number of lattice points on spheres of any radius, this component of our method exploits a comparable uniform but essentially elementary bound for the number of lattice points on hyperplanes. This is the basis for our lower bound condition on the upper Minkowski dimension in Theorem 4. Denoting the upper Minkowski dimension of F by e F, the actual exponent in the lower bound of distinct volumes is, however, weaker. The value of 1 n 1 e F ε ( ε) is rather far from 1 e F ε ( ε), which is the exponent if the sets obtained from the procedure of normalization in 5 [ibid.] are both Salem and s adaptable for any s > e F. It evidently only approaches this value when e F is close to n. It does not, however, seem implausible to believe that it could be improved by finding good estimates for the density of lattice points on the determinantal varieties that actually appear in the diophantine problem. Extending such estimates to the highly singular algebraic sets encountered with the volume invariant could improve some of our results. It does not seem too unreasonable to believe that the pioneering work of [BH-BS] can be pushed further to do ust that. The analogy between Salem sets and thickness clearly deserves to be understood more completely. It also ustifies trying to understand how frequently the latter property is actually encountered when working with self similar sets. In a separate article [EL-3], we explore this question, and show that thickness is an asymptotically generic property when confined to classes of compatible self similar sets that Strichartz has called non overlapping (see [St]). A useful problem for further work, it seems to us, would be to find more intrinsic characterizations of the thickness property, connecting it to underlying symmetries as done in 4.3 for the permutation group. This should also help identify additional examples of thick self similar subsets of some Z n that occur naturally in combinatorial geometry or number theory. Our proof of the asymptotic genericity property suggests to us that there should be many such examples. 2 Statements of quantitative results, Technical preliminaries, and a Multivariate Tauberian theorem 2.1 Basic definitions and Statements of results We first recall the basic notion of a compatible self-similar subset of R n from [EL-1]. We fix in the sequel (E, q) a Euclidean space, dim R E = n, with standard Euclidean norm = q 1/2, and bilinear form B(x, y) = x, y the usual scalar product. Definition 1. Let T i (i I) be a set of orthogonal linear transformations of (E, q) that pairwise commute. A family f i = c i T i + b i (i I) of similarities of E is then said to be compatible. The constants c i are the scale factors of the similarities. Definition 2. Let F be a countable discrete subset of E. Define the upper Minkowski dimension of F by: ln [ #F(R) ] u dim M F := lim [0, ] where F(R) := F B(R) R. R ln R 6

We say that F has finite upper Minkowski dimension whenever the limit is finite. Remark: In [EL-1], [EL-2], we used the term exponent but in retrospect, this seems to have been unnecessarily confusing. In this case the zeta function of F is a series summed over F := F {0} ζ(f; s) := m s m F that converges absolutely in the halfplane σ(:= Rs) > u dim M F, and s = u dim M F is its abscissa of convergence. Definition 3. A countably infinite discrete subset F E is said to be a compatible self-similar set if these two properties are satisfied: 1. 0 < u dim M F <. 2. There exists a finite compatible set f = {f i } r i=1 of affine similarities such that each scale factor c i > 1 and 1 F r i=1f i (F) and f i (F) f i (F) is finite if i i. (4) For convenience, and to be consistent with the notation used throughout [ibid.], we will set e F := u dim M F. There is a simple analytic characterization of e F (see Theorem 2 of [EL-1]) as follows: (3) e F is the unique real solution of the equation r =1 c s = 1. (5) In the rest of this article, we use the notation F to denote a compatible discrete self-similar subset of R n. We also set D F := e F + 2. (6) Statements of our quantitative theorems We prove six explicit lower bound results, each of an asymptotic nature in a large parameter x, for the metric invariants defined in the Introduction. Despite the variety of results, each can be stated very concisely. For any x > 0, we first define (see Definition 2) Dis F (x) := #{ m 1 m 2 : m 1, m 2 F(x)}, (7) with the understanding that Dis F (x) counts each distance exactly once. We prove in Section 3.1 the following. 1 The notation F G means that (F \ G) (G \ F ) is a finite set. 7

Theorem 1. Assume F is a compatible self similar set inside Z n (n 2) with upper Minkowski dimension e F > n 2. Then for any sufficiently small ε > 0, n 2 1 ε e Dis F (x) ε [#F(x)] F as x. (8) Note: For simplicity, we will simply write for sufficiently small ε to mean that ε should be smaller than a certain linear expression involving e F and n (that depends upon the particular problem under consideration) in order that the exponent of a monomial in x is positive. It is also natural to want an extension of Theorem 1 to angles formed by pairs of points with a fixed origin. To this end, we define for m 1, m 2 F : θ(m 1, m 2 ) = the angle formed between m 1 m 1 and m 2 m 2, (9) where we think of the two unit vectors as belonging to the unit sphere with center the origin 0. We then define We prove in Section 3.2 the following. Ang(x) = #{cos ( θ(m 1, m 2 ) ) : m 1, m 2 F(x)}. (10) Theorem 2. Assume F is a compatible self similar set inside Z n with upper Minkowski dimension e F > n 2. If n 4, then for any sufficiently small ε > 0: n 2 1 ε e Ang(x) ε [#F(x)] F (x 1). (11) We prove in Section 3.3 a natural extension of Theorem 1 to rooted configurations. Define Root k,f (x) := # {( m 1 m k+1,..., m k m k+1 ) : m F(x) and m m k+1 k }. (12) Theorem 3. Assume F is a compatible self similar inside Z n (n 2) with upper Minkowski dimension e F > n 2. Then for sufficiently small ε, n 2 k[1 ] ε e Root k,f (x) ε [#F(x)] F as x. (13) An immediate consequence of Theorem 3 is an extension to arbitrary k point configurations. Define Config k,f (x) {( ) := # m 1 m k+1,..., m k m k+1, m 1 m k,..., m k 1 m k,..., m 1 m 2 : } m F(x) and m i m i. 8

Corollary 1. Assume F is a compatible self similar inside Z n (n 2) with upper Minkowski dimension e F > n 2. Then for sufficiently small ε, Config k,f (x) ε [#F(x)] k(k+1) 2 [1 n 2 e F ] ε as x. (14) To state the last three results we first define for any vector m = (m 1,..., m n+1 ) F n+1 and n simplex Σ n (m) (see Introduction), and set Σ n (m) = det(m 1 m n+1,..., m n m n+1 ), V ol n,f (x) := #{ Σ n (m) : m F(x) n+1 } (15) to denote the number of distinct volumes whose endpoints belong to F(x) n+1. We define in Section 4.1 what we mean by thickness of a self similar set (see Definition 4), and prove in Section 5 the following three results. Theorem 4. Assume that F Z n is thick and compatible, and e F > n 1. Then for sufficiently small ε > 0: V ol n,f (x) ε [ #F(x) ] 1 n 1 e F ε as x +. (16) In Section 4.3, we show that Pascal s triangle mod any prime p, denoted P as(p), is a thick subset of Z 2. It is also known that its upper Minkowski dimension e p is given by e p = ln(p(p + 1)/2)/ ln p. Since e p > 1, Theorem 4 applies since it follows directly from the construction of the set that P as(p) is compatible. So, we conclude as follows. Theorem 5. For any prime p and sufficiently small ε : V ol 2,P as(p) (x) ε [ #P as(p)(x) ] 1 1 ep ε as x. (17) Since e p 2 when p, we also conclude that (17) implies V ol 2,P as(p) (x) ε [ #P as(p)(x) ] 1 2 ε as x and for all p ε 1. (18) It appears that both of these lower bounds describe new features about the fractal like sets P as(p). Moreover, it is perhaps surprising to discover that there is a simple monomial lower bound for the number of distinct areas of triangles, as counted by V ol 2,P as(p) (x), whose exponent is uniform over all sufficiently large p in the sense that for any ε > 0 (and sufficiently small), we can choose this exponent to equal 1 2 ε whenever p p ε. In Section 4.3, we also show that the Pascal pyramid mod p, denoted M n,p, is thick if p n. These sets are natural generalisations of P as(p) that use multinomial coefficients instead of binomial coefficients. The upper Minkowski dimension of these sets we denote by e n,p. We showed in [EL-1] that ( (p+n 1 ) ) ln p 1 e n,p =. ln p 9

Theorem 6. If ( ) p + n 1 > p n 1, (19) p 1 then for any sufficiently small ε > 0: V ol Mn,p (x) ε [ #Mn,p (x) ] 1 n 1 en,p ε as x +. (20) Remark: It easy to see that: 1. n = 2 implies (19) holds for any p; 2. n = 3 implies (19) holds for any p 3; 3. n 4 implies (19) holds for any p n!. 2.2 Two technical preliminaries 2.2.1 Properties of Dirichlet polynomials with complex coefficients Let c = (c 1,..., c r ) (1, ) r and γ = (γ 1,..., γ r ) C r. We define a Dirichlet polynomial with complex coefficients r Λ(c, γ, s) = γ c s 1. =1 We first recall from (5) that e F is the unique real solution of the equation r =1 c s = 1. It follows that the line {σ = e F } is the rightmost vertical line that can first contain a root of Λ(c, 1, s), in the sense that there are no roots in the halfplane σ > e F. The following Lemma 1 gives us useful information about the set of roots of Λ(c, γ, s) lying in some neighborhood of the vertical line {σ = e F } when γ (S 1 ) r. This will be used in 2.2.2. Parts 1-3 of the Lemma were proved in [LF1] (ch. 3) for γ (0, ) r. The proofs extend without significant difficulty to vectors γ (S 1 ) r (and even more generally to γ (C ) r ). Parts 4, 5 of Lemma 1 extend the discussion in [ibid.] (see Theorems 3.25 and 3.26) since this only seems to apply to γ N r. The reader may also find it helpful to consult the article [FRV] that adds additional details to those given in [op.cit.]. As a result, the proofs of Parts 4,5 in Lemma 1 in the case γ (S 1 ) r appear to be both new and needed. The essential idea is that we can reduce this case to that of γ = 1 by a translation of s (along the imaginary axis). We first recall from [LF1], [LF2], that there are two distinct cases latticelike case and non latticelike case. This depends solely upon the behavior of the roots of the polynomial Λ(c, 1, s). 10

The latticelike case corresponds to the case where the subgroup of =1 Z ln c of R generated by ln c 1,..., ln c r is discrete. The non latticelike case corresponds to the possibility where this group is dense in R. Lemma 1. Assume that γ (S 1 ) r. The following properties are then satisfied by the roots of Λ(c, γ, s). 1. Each root lies in the halfplane {σ e F }. 2. Any root on the line {σ = e F } is a simple zero. 3. Defining the number of zeroes up to height T we have the estimate N c,γ (T ) = {s C; Λ(c; γ; s) = 0 and Is T } N c,γ (T ) = O(T ) uniformly in T 1 (where zeroes are counted according to multiplicity); 4. If at least one root of Λ(c, γ, s) lies on the line σ = e F, then there exist µ (0, 1), ω = ω(c) 0, a = a(c; γ) R, and a curve G c,γ : τ (g c,γ (τ), τ), where g c,γ (τ) : R (e F µ, e F ) is a continuous function, satisfying the following three properties: (a) In the non latticelike case, ω = 0. In the latticelike case we can set g c,γ (τ) = σ 0 for any fixed σ 0 (e F µ, e F ). (b) The zeroes of Λ(c; γ; s) on the line {σ = e F } are contained in the set {e F + i(a + lω)} l Z. (c) Λ(c; γ; s) satisfies the lower bound Λ(c, γ, s) Gc,γ (τ) 1 for all τ, and all its zeroes to the right of G c;γ are simple. The curve G c,γ is called a screen for the Dirichlet polynomial Λ(c, γ, s). 5. Defining the window W c,γ := {s = σ + iτ; σ g c,γ (τ)} for the screen G c,γ the following two properties are satisfied: (a) The derivatives of Λ(c, γ, s) satisfy the property: Λ (c, γ, ρ) 1 uniformly over the zeroes ρ of Λ in W c,γ ; (b) There exist two monotonic unbounded sequences {T n } n 1 (0, ) and {T n } n 1 (, 0) such that: Λ(c, γ, σ + it ±n ) 1 uniformly in n N and σ g c,γ (T ±n ). 11

Remark: In particular, in the non latticelike case, if there is a pole on the line {σ = e F } then it is unique. In the latticelike case, the lower bounds in parts 4 (c) and 5 (b) are satisfied even when there is no zero of Λ(c, γ, s) on the line σ = e F. The other parts are all vacuously satisfied in this event. Proof : The first point concerns the proof of Part 3. This is an immediate consequence of Theorem 3.6 of [LF1]. Proof of Part 1: r =1 c σ Let s = σ + iτ C. If σ > e F, then r =1 γ c s < r =1 c e F = 1. It follows that Λ(c; γ; s) 0. r =1 γ c s = Proof of Part 2: Assume that s = e F + iτ is a zero of Λ(c; γ; s) on the line {σ = e F }. It follows that 1 = r =1 γ c s r =1 γ c s = r =1 c e F = 1. Thus, there exist positive numbers µ 1,..., µ r > 0 and a real number θ such that γ c s = µ e iθ for all = 1,..., r. We deduce that d r r r ds Λ(c; γ; s) = γ c s ln c = µ e iθ ln c = µ ln c > 0. =1 Therefore, s must be a simple root of Λ(c; γ; s). Proof of Parts 4 and 5: Assume that Λ(c; γ; s) has at least one root s 0 = e F + iτ 0 on the line σ = e F. It follows that 1 = r =1 γ c s 0 r =1 γ c s 0 = r =1 c e F = 1. As in the preceding step, µ 1,..., µ r > 0 and a real number θ exist so that Since each γ = 1, (21) implies =1 γ c s 0 = µ e iθ for all = 1,..., r. (21) =1 µ = c e F and γ c iτ 0 = e iθ for all = 1,..., r. Substituting these identities in the equation Λ(c; γ; s 0 ) = 0 we get: Thus, have 1 = r =1 γ c s 0 = r =1 c e F γ c iτ 0 = e iθ r =1 c e F = e iθ. µ = c e F and γ c iτ 0 = 1 for all = 1,..., r. It follows that for any s C, we Λ(c; γ; s) = 0 r =1 γ c s = 1 r =1 c (s iτ 0) = 1 Λ(c; 1; s iτ 0 ) = 0, As a result, we see that the general case reduces to the particular case where γ = 1 for all, to which Theorems 3.25, 3.26 of [LF1] do apply. This completes the proof of the lemma. 2.2.2 Twisted fractal zeta functions Before beginning the discussion here, we first introduce some needed 12

Notations: We set N 0 = N {0}, and for any α = (α 1,..., α n ) N n 0, λ = (λ 1,..., λ n ) C n, we define α = α i and λ α := λ α. i To define the twisted fractal zeta functions, we must first fix an orthonormal basis B = {g 1,... g n } of E C, the complexification of E, with respect to which each T is diagonalizable. Each element m of F is then written in the basis B as a linear combination m = m g. In these coordinates, we define the (a priori formal) twisted fractal zeta functions ζ F (α; s) := m F m α m s (s = σ + iτ). (22) The proof of the following Lemma then becomes a routine application of the idea from [EL-1] and Lemma 1. In addition, one sees quite clearly, how knowledge about the zeroes of the Dirichlet polynomial Λ(c, λ α, s) translates into information about the poles of ζ F (α, s). Lemma 2. Let α = (α 1,..., α n ) N n 0. ζ F (α, s): The following properties are satisfied for any 1. s ζ F (α; s) converges absolutely in {σ > e F + α }, has a meromorphic continuation with moderate growth 2 to the complex plane C and its polar locus is a subset of α N n 0 k N { s + α k : r =1 2. For any η (0, 1), the function s ζ F (α; s) := } λ α c s = 1. ( ) r =1 λα c α s 1 ζ F (α; s) holomorphic on the halfplane {σ > e F + α η} where it satisfies the estimate ζ F (α; σ + iτ) 1 + τ η. (23) 3. If e F + α is a pole of ζ F (α; s), then it is a simple pole and λ α = 1 for all = 1,..., r. 4. If ζ F (α; s) has at least one pole in the line σ = e F + α, then there exist µ (0, 1), a = a(c; α) R, ω = ω(c) 0, δ = δ(c, α, µ) (0, 1) and a screen G c,α : τ (g c,α (τ), τ) = (σ, τ), where g c,γ : R (e F + α µ, e F + α ), such that: (a) The poles of ζ F (α; s) on the line σ = e F + α are contained in the set {e F + α + i(a + lω); l Z}. Moreover, ω = 0 if F is non latticelike; 2 A meromorphic function F (s) with polar locus P has moderate growth on a domain D C if there exists a, b > 0 such that δ > 0 and s {d(s, P D) δ}, F (σ + iτ) σ,δ 1 + τ a σ +b (see [EL-1]). is 13

(b) The poles ρ of ζ F (α; s) in the window W c,γ := {s = σ + iτ; σ g c,α (τ)} are simple and the following estimate is satisfied: Res s=ρ ζ F (α, s) 1 + Iρ δ ; (c) ζ F (α, s) has no pole on the screen G c,α ; (d) ζ F (α, s) 1 + Is δ uniformly in s G c,α ; (e) There exist two monotonic unbounded sequences of real numbers {T n } n 1 (0, ) and {T n } n 1 (, 0) such that ζ F (α; σ + it ±n ) 1 + T ±n δ uniformly in n N and σ g c,α (T ±n ). 5. In the latticelike case Parts 4 (a) - (e) remain true even if ζ F (α, s) has no pole on the line σ = e F + α. Moreover, the screen in this event can be chosen to be a vertical line to the left of {σ = e F + α }; 6. In the non latticelike case, set {0 < λ 1 < λ 2 < } to denote the set of distinct values of the function m F m, and define b k (α) := m α for all k 1. {m F ; m i =λ k } Then there exists a sequence {a k (α)} k 1 (0, ) such that (a) b k (α) a k (α) for all k 1; (b) the Dirichlet series s a k (α) k=1 λ s converges absolutely in the half-plane {σ > k e F + α } and has a meromorphic continuation to an open domain containing {σ e F + α }. This meromorphic function has exactly one pole on s = e F + α which is simple. Proof of Lemma 2: Part (1) follows from Theorem 1 of [EL-1]. Part 4 follows from Part 2 and Lemma 1 Parts 4 and 5. Part 5 follows from Part 2, Lemma 1 Part 4(a), and the Remark following the statement of that Lemma. Proof of Part 2: We first remark (see Footnote 1) that for σ > e F + α : ζ F (α; s) = n k=1 mα k k m s = n k=1 m, g k α k r n k=1 m s f (m), g k α k f m F m F =1 (m) s m F r n ( k=1 c T (m + u ), g k ) α k r n ( k=1 c m + u, T c T (m + u ) s (g k) ) α k c T (m + u ) s =1 m F r =1 m F n k=1 ( c λ k, m + u, g k ) α k c T (m + u ) s =1 m F r =1 m F c α s λ α (m + u ) α m + u s 14

It follows that for σ > e F + α : ( r ) ζ F (α; s) = λ α c α s 1 ζ F (α; s) =1 r =1 c α s λ α m F [ m α m s (m + u ) α ] m + u s. (24) Now let K be any compact subset of R. We then observe that the following estimate is uniform over all m F such that m 1 and all s = σ + iτ C such that σ K: m α m s (m + u ) α m + u s u K (1 + τ ). (25) m σ 1 We conclude that s ζ F (α; s) has a holomorphic continuation with moderate growth to the set {σ > e F + α 1} on which ζ F (α; σ + iτ) 1 + τ. By using in addition the classical Phragmén-Lindelöf principle (see [Te], Th. 16, p. 123), we get the growth estimate (23) for ζ F (α, s). Proof of Part 3: Assume that s = e F + α is a pole of ζ F (α; s). We conclude from Part (2) that r λ α c e F = 1. (26) =1 Applying Theorem 2 [EL-1], we deduce that 1 = r =1 λ α c e F r =1 λ α c e F = r =1 c e F = 1. It follows that λ α /λ α 1 (0, ) for all = 1,..., r, and hence λ α = λ α 1. Relation (26) then implies that λ α 1 = λ α 1 We deduce that λ α = 1 for all. ( r =1 c e F ) = r =1 λ α c e F = 1. It follows that ( r d λ α c α s 1) ds =1 s=e F + α = r =1 c e F ln c 0, (27) Combining this with the fact that ζ F (α; s) is analytic at s = e F + α then implies that e F + α is a simple pole of ζ F (α; s). Proof of (6): Assume we are in the nonlatticelike case. Set a k (α) := {m F ; m i =λ k } m α for all k 1. Then it s easy to see that for σ > e F + α, we have G(s) := k=1 a k (α) λ s k = m F m α m s = m F 1 m s α = ζ(f, s α ), 15

where ζ(f, s) = m F 1 m s. We conclude by applying Theorems 1 and 2 of [EL-1]. This completes the proof of Lemma 2. Remark : The difference between the compatible self similar case and non compatible case is most clearly seen in the nature of the Dirichlet polynomial Λ(c, γ, s), appearing in Part 1 of Lemma 2. In the compatible case, each γ = 1. This however need not occur in general. As a result, the main difficulty is to extend the properties of the roots of Λ(c, γ, s) proved in Lemma 1 to any complex vector γ. 2.3 A multivariate Tauberian theorem The main result of this subsection is a multivariate Tauberian theorem (see Theorem 8) that suffices for our purposes, though it is hardly the most general result that can be proved. We will apply it to prove each of the theorems stated in 2.1. Since we are unable to find anything remotely comparable in the literature, we believe this result is new. We start with a function f : N k R + and an unbounded increasing sequence of positive numbers 0 < λ 1 < λ 2 <. We then formally define a multivariate Dirichlet series by setting Z(f, s) := f(n 1,..., n k ) λ s 1 n 1 λ s (s = (s k 1,..., s k ) C k ), (28) nk n 1,...,n k 1 and the averages of the coefficients of Z(f, s) : A(f, x) := f(n 1,..., n k ) (x = (x 1,..., x k ) [1, ) k ). (29) λ n1 x 1,...,λ nk x k We next impose conditions about Z(f, s) that are satisfied by each of the series in this article. The properties specified below combine all the features proved in [ibid.] for both the lattice and non latticelike cases. Hypotheses assumed satisfied by Z(f, s). (I) There exist Dirichlet series Z b, (s), each of the form Z b, (s) = n=1 a b, (n) λ s n and e b, > 0 such that Z b, is absolutely convergent when σ > e b,. (II) There exist complex numbers d b C such that in the domain {σ > D}, where D = max b, e b,, B k Z(f, s) = Z b, (s ). (30) d b b=1 =1 Moreover we assume that there exist µ > 0, δ (0, 1) and ω 0 such that for any b {1,..., B} and for any {1,..., k}, the Dirichlet series Z b, (s) verifies one of the two following assumptions (III) or (IV):, 16

(III)There exists a continuous function g b, properties are satisfied : : R (e b, µ, e b, ) so that the following 1. Z b, (s) has a meromorphic continuation to an open neighborhood of the set W b, := {s C : σ g b, (τ)} ; 2. Any pole ρ of Z b, (s) in W b, is simple and Res s=ρ Z b, (s) 1 + Iρ δ ; 3. There exists φ b, R such that the poles of Z b, (s) on the line σ = e b, are elements of {e b, + i(φ b, + lω); l Z}; 4. There is no pole on the curve G b, := {g b, (τ) + iτ : τ R}(= W b, ); and N b, (t) := number of poles ρ of Z b, (s) in W b, such that Iρ t satisfies the bound: 5. Z b, (s) 1 + τ δ uniformly in s G b, ; N b, (t) = O(t) as t ; 6. There exist two unbounded monotonic sequences {T n } n 1 (0, ) and {T n } n 1 (, 0) such that Z b, (σ + it ±n ) 1 + T ±n δ uniformly in n N and σ g b, (T ±n ). or (IV) There exists a sequence of nonnegative number {a b, (n)} n 1 such that: 1. a b, (n) a b, (n) for all n 1; 2. The Dirichlet series s Z b, (s) := a b, (n) n=1 λ converges absolutely in the open halfplane {σ > e b, } and has a meromorphic continuation to an open domain containing s n the closed half-plane {σ e b, } with no pole on the line σ = e b, ; 3. The Dirichlet series s Zb, (s) := a b, (n) n=1 λ converges absolutely in the open halfplane {σ > e b, } and has a meromorphic continuation to an open domain containing s n {σ e b, } with a single pole that is simple at s = e b,. Remark: For our purposes, the individual factors Z b, will all be of the form: There exists a monomial α b, such that Z b, (s) = ζ F (α b,, s). As a result, the needed properties are all, indeed, satisfied by the discussion in 2.2.2. Denoting Z(f, s) by Z for simpliciity, and using the e b, we first define the set of points S(Z) := {e(b) := (e b,1,..., e b,k ) : b = 1,..., B}, 17

weights e(b) := e b,, and maximum weight ω(z) := sup{ e : e S(Z)}. With this data, we then define a polyhedron and a particular subset of its vertex set as follows. 1. S 0 (Z) := {e S(Z); e = ω(z)}; 2. Γ(Z) = convex hull{e S 0 (Z); e = ω(z)}; 3. V(Z) = {v : v S 0 (Z) and v is a vertex of Γ(Z)}. (31) We use the preceding properties of Z(f, s) to describe explicitly the following weighted average : k ( H(f, x) := f(n 1,..., n k ) 1 λ ) n. (32) x λ n1 x 1,...,λ nk x k =1 It is clear that H(f, x) A(f, x). An application of Perron s weighted formula (see [I]) tells us that for any ξ > max{e b, }, H(f, x) = 1 (2πi) k ξ+i ξ i ξ+i... ξ i Z(f, s) k =1 x s s (s + 1) ds 1... ds k (33) Applying (30), we first express the left side more explicitly as follows: H(f, x) = B k d b b=1 =1 H b, (x ) where H b, (x ) := 1 2πi ξ+i ξ i Z b, (s ) x s s (s + 1) ds. (34) A consequence of this application is the construction of the following quantity which will play the role of the expected dominant term for H(f, x). For each e V(Z), set ψ e (y) = {b:e(b)=e} l Z k c e (l; b) e i φ(b)+ωl,y where, for all b φ(b) = (φ b,1,..., φ b,k ) and for l = (l 1,..., l k ), c e (l; b) := d b k =1 Res s =e +i(φ b, +ωl ) Z b, k =1 (e + i(φ b, + ωl ))(e + 1 + i(φ b, + ωl )). (y = (y 1,..., y k )), (35) We note that (III) part 2 and (IV) part 2 of the Hypotheses imply that each ψ e is an absolutely convergent series when F is latticelike. In addition, the following property is satisfied: c e (l; b) 2 <. (36) {b:e(b)=e} l Z k 18

If, however, F is non latticelike, then ψ e is a finite sum (over the b so that e(b) = e) of the products of residues at e b,. Our first result is as follows. Theorem 7. H(f, x) = e V(Z) x e ψ e (ln x 1,..., ln x k ) + o e V(Z) In order to prove Theorem 7 we need the following lemma: Lemma 3. Let b {1,..., B}, {1,..., k} and ξ > e b,. The function H b, (x ) defined in (34) satisfies (as x ) : x e + O e S(Z)\V(Z) x e (37) H b, (f, x ) = ( ) x e b,+i(φ b, +ωl ) Res s =e b, +i(φ b, +ωl ) Zb, (e b, + i(φ b, + ωl ))(1 + e b, + i(φ b, + ωl )) +o(xe b, ). l Z (38) In case (III) and ω = 0 in the non latticelike case resp. in case (IV), the above sum over l is replaced by a single residue at s = e b, + iφ b, resp. 0. Proof of Lemma 3: First case: we assume that assumption (III) holds. For each n, we transport the segment {(ξ, τ ) : T n τ T n } to the left and stop along the curve C(b,, n) := {(S b, (τ ), τ ) : T n τ T n }. Along the way we encounter the possible pole of Z b, (s ) at s = e b, + i(φ b, + ωl ) as well as O(T n + T n ) other poles ρ b, (r) W b, {σ = e b, }. Denoting the upper resp. lower horizontal segments τ = T n resp. τ = T n as W (+n) resp. W ( n) it is clear that if ω 0 then 1 2πi ξ+itn T n ω l Tn ω + + x s Z b, (s ) ξ+it n s (s + 1) ds = ( ) x e b,+i(φ b, +ωl ) Res s =e b, +i(φ b, +ωl ) Zb, (e b, + i(φ b, + ωl ))(1 + e b, + i(φ b, + ωl )) T n I ρ b, (r) T n Res s =ρ b, (r) W (+n) W ( n) ( ) x ρ b,(r) Zb, ρ b, (r)(ρ b, (r) + 1) x s Z b, (s ) s (s + 1) ds + C(b,,n) x s Z b, (s ) s (s + 1) ds. (39) If ω = 0 (non latticelike case), then the above sum over l is replaced by a single residue at s = e b, + iφ b,. No change is needed in the three other summands. 19

Applying parts 5 and 6 of (III), it is clear that for any x : lim n + lim n + Z b, (s ) s (s + 1) ds = 0 (40) x s Z b, (s ) s (s + 1) ds x s = Z b, (s ) s (s + 1) ds exists. W (+T n) W ( T n) C(,β,n) Part 2 of (III) implies x s C (β) l Res s =e b, +i(φ b, +ωl ) ( Zb, ) x e b,+i(φ b, +ωl ) (e b, + i(φ b, + ωl ))(1 + e b, + i(φ b, + ωl )) converges absolutely. Moreover, since σ Cb, < e b, for each s C b, an easy application of Lebesgue dominated convergence theorem tells us that x s Z b, (s ) C b, s (s + 1) = o(xe b, ) as x. (41) The last point to address is the behavior in x of the limit as n of T n I ρ b, (r) T n Res s =ρ b, (r) ( Zb, ) x ρ b,(r) ρ b, (r)(ρ b, (r) + 1). We first show that the series obtained by replacing the monomial in x by the constant 1 converges absolutely. By choosing µ small enough, we can assume that e b, µ > 0. Set γ = 1 2 (e b, µ). Part 4 of (III) implies that we have uniformly in T 1: ( ) Res s =ρ b, (r) Zb, 1 T ρ b, (r)(ρ b, (r) + 1) (γ + Iρ b, (r) ) 2 δ = (γ + t) 2+δ dn b, (t) 0 Im ρ b, (r) T Iρ b, (r) T = (γ + T ) 2+δ N b, (T n ) + (2 δ) T 1+δ + γ t 2+δ dt 1. T 0 (γ + T ) 3+δ N b, (t) dt As a result, by using the fact that R ρ b, (r) < e b, for each r, an application of the Lebesgue dominated convergence theorem (using the Stieltes integral with respect to dn b, (t)) then shows: lim Res s =ρ n b, (r) I ρ b, (r) T n ( Zb, ) x ρ b,(r) ρ b, (r)(ρ b, (r) + 1) = ρ b, (r) ( ) x ρ b,(r) Res s =ρ b, (r) Zb, ρ b, (r)(ρ b, (r) + 1) = o(x e b, ) ( as x ). (42) 20

We conclude that (38) holds if the hypotheses in (III) are all satisfied. Second case: we assume that (IV) holds. We will prove first that A b (x ) := a b, (n) = o(x eb, ) λ n x as x. (43) Proof of (43): We will deduce this estimate from Delange s tauberian theorem [Del] by using, in addition, an argument in [L] (see Theorem 1 pg. 311). Set Z b,,con (s) := a b, (n) n=1 λ. It s easy to see that Z s n b,,con (s) := Z b,,con (s). We deduce that the series verifies the same assumptions as Z b, (s). It follows that the following two Dirichlet series Z b,,r (s) := Z b,,i (s) := a b, (n) R (a b,(n)) = Zb, (s) 1 2 (Z b,(s) + Z b,,con (s)) and ; n=1 λ s n a b, (n) I (a b,(n)) = Zb, (s) 1 2i (Z b,(s) Z b,,con (s)) n=1 λ s n have non negative coefficients, and also satisfy the same assumptions as Z b, (s). Moreover, each series has the same residue at the pole s = e b,. We next define R := Res s=eb, Z b, (s)/e b,. Delange s tauberian theorem implies then that a b, (n) R (a b,(n)) = Rx eb, λ n x a b, (n) I (a b,(n)) = Rx eb, λ n x λ n x a b, (n) = Rxeb, + o(x e b, ), + o(x e b, ), + o(x e b, ). It follows that R (a b, (n)) = o(x eb, ) and I (a b, (n)) = o(x eb, ). λ n x λ n x This completes the proof of (43). We now conclude from (43) that H b, (f, x ) = ( a b, (n) 1 λ ) n x λ n x = x 1 x 0 A b (t) dt = o(x e b, ) as x. Thus, (38) is also satisfied in case (IV). This finishes the proof of Lemma 3. 21

Proof of Theorem 7: have It follows from (33) and (38) that when x (+,..., + ), we H(f, x) B = = d b b=1 =1 B { d b b=1 k ( l Z ( ) x e b,+i(φ b, +ωl ) ) Res s =e b, +i(φ b, +ωl ) Zb, (e b, + i(φ b, + ωl ))(1 + e b, + i(φ b, + ωl )) + o(xe b, ) (l 1,...,l k ) Z k x e(b)+iφ(b) ( ( ) ) k Res s =e b, +i(φ b, +ωl ) Zb, (e b, + i(φ b, + ωl ))(1 + e b, + i(φ b, + ωl )) xiωl =1 Rearranging the terms and making evident simplifications shows that H(f, x) satisfies the asserted asymptotic when x (+,..., + ). In particular, the factor of x e(b)+iφ(b) is an absolutely convergent Fourier series in the latticelike case. In the non latticelike case, the factor of x e(b)+iφ(b) is a linear combination (over b such that e(b) = e) of d b multiplied by an iterated residue at s = e(b). Each such factor is denoted by ψ e. The next point applies this to derive a nontrivial lower bound for A(f, x) := A(f, (x,..., x)) = f(n 1,..., n k ). (44) λ n1 x,...,λ nk x This is possible when, for any vertex e V(Z), the iterated residue of Z(= Z(f, s) at the point e is non vanishing. We denote/define this quantity by Reg Z(f, e) := B d b b=1 =1 k Res s =e b, Z b, (s ). (45) More precisely, we will derive from Theorem 7 the following Tauberian theorem. Theorem 8. Assume that for any vertex e V(Z) (Z = Z(f, s)) Then, for any ε > 0, there exists C(ε) > 0 such that Reg Z(f, e) 0. (46) A(f, x) C(ε) x ω(z) ε uniformly in x 1. (47) } + o(x e(b) ) Remark: In the following, we abbreviate (47) by writing A(f, x) ε x ω(z) ε when x. Proof : First we remark that we can assume without loss of generality that for all b = 1,..., B and all = 1,..., k, we have φ b, = 0 or φ b, Qω. (48) Given e V(Z) such that (46) holds for some l Z k, it follows that the series ψ e (y) defined in (35) is a nonzero absolutely convergent Fourier series. Using ψ e, we now construct a 22

nonzero almost periodic function in a single variable with positive mean for any sufficiently small ε > 0 We first choose any ε. Next, we remark that there exists θ = (θ 1,..., θ k ) [1, 1 + ε) k such that e, θ e, θ e e S(Z) and (49) φ(b) + ωl, θ 0 ifl Z k \ {(0,..., 0)} or φ(b) (0,..., 0) It suffices to choose a vector θ each of whose coordinates is transcendental over Q ( ω, {φ(b)} b B ) and belongs to [1, 1+ε), such that the vector lies in the open subset defined by the following conditions: e e, θ 0 for all e e S(Z). Clearly, such θ exist. It then follows that there exists a unique e V(Z) such that e, θ > e, θ e S(Z) \ {e}. (50) Setting x = x θ = 1,..., k in Theorem 7, we conclude: A(f, x 1+ε ) A(f, x θ 1,..., x θ k ) H(f, x θ 1,..., x θ k ) = x e,θ [ψ e (θ 1 ln x,..., θ k ln x) + o (1)]. (51) We now define the function of a single real variable y: ψ e,θ (y) := ψ e (θ 1 y,..., θ k y) = c e (l; b) e i φ(b)+ωl,θ y. {b:e(b)=e} l Z k Denoting the distinct values φ(b) + ωl, θ as (η n ) n Z, it follows that η n η m if n m. As a result, when we rearrange the terms and write ψ e,θ as a series summed over n ψ e,θ (y) = n Z C n e iyηn. the algebraic independence property in (49) insures three essential properties: 1. There exists a unique integer n 0 such that C n0 = {b; e(b)=e φ(b)=0} d b k =1 Res s =e Z b, k =1 e (e + 1) = Reg Z(f, e) k =1 e (e + 1) 0. 2. The series ψ e,θ (y) converges absolutely and is nonzero. 3. The function ψ e,θ 2 has a non zero mean value that does not depend upon θ. 23