Minkowski dimension and explicit tube formulas for p -adic fractal strings
aa r X i v : . [ m a t h - ph ] O c t Minkowski dimension and explicit tube formulas for p -adicfractal strings Michel L. Lapidus, L˜u’ H`ung, and Machiel van Frankenhuijsen
Abstract.
The theory of complex dimensions describes the oscillations in thegeometry (spectra and dynamics) of fractal strings. Such geometric oscillationscan be seen most clearly in the explicit volume formula for the tubular neigh-borhoods of a p -adic fractal string L p , expressed in terms of the underlyingcomplex dimensions. The general fractal tube formula obtained in this paperis illustrated by several examples, including the nonarchimedean Cantor andEuler strings. Moreover, we show that the Minkowski dimension of a p -adicfractal string coincides with the abscissa of convergence of the geometric zetafunction associated with the string, as well as with the asymptotic growthrate of the corresponding geometric counting function. The proof of this newresult can be applied to both real and p -adic fractal strings and hence, yieldsa unifying explanation of a key result in the theory of complex dimensions forfractal strings, even in the archimedean (or real) case. Contents
1. Introduction 22. Nonarchimedean Fractal Strings 43. The Geometric Zeta Function 104. Volume of Thin Inner Tubes 125. Minkowski Dimension 176. Explicit Tube Formulas for p -adic Fractal Strings 227. Possible Extensions 278. Epilogue 28Acknowledgments 30 Mathematics Subject Classification.
Primary 11M41, 26E30, 28A12, 32P05, 37P20;Secondary 11M06, 11K41, 30G06, 46S10, 47S10, 81Q65.
Key words and phrases.
Fractal geometry, p -adic analysis, p -adic fractal strings, zeta func-tions, complex dimensions, Minkowski dimension, fractal tubes formulas, p -adic self-similar strings,Cantor, Euler and Fibonacci strings.The work of the first author (MLL) was partially supported by the US National ScienceFoundation (NSF) under the research grants DMS-0707524 and DMS-1107750, and by the Institutdes Hautes Etudes Scientifiques (IHES) in Paris/Bures-sur-Yvette, France, where the first authorwas a visiting professor while part of this work was completed, as well as by the Burton JonesEndowed Chair in Pure Mathematics (of which MLL was the chair holder at the University ofCalifornia, Riverside, during the completion of this paper).The research of the second author (LH) was partially supported by the Trustees’ ScholarlyEndeavor Program at Hawai‘i Pacific University. References 30
Nature is an infinite sphere of which the center is everywhereand the circumference nowhere.
Blaise Pascal (1623–1662)
1. Introduction
An ordinary real (or archimedean) fractal string is a bounded open subsetof the real line, with fractal boundary. It provides a complementary perspectiveto the notion of a self-similar fractal, in the sense that every self-similar stringdetermines a self-similar set in R , viewed as the boundary of the string. Moreover,it is noteworthy that the geometric zeta function of a fractal string determines thefractal (i.e., Minkowski) dimension of the corresponding fractal set. Following theexamples of the a -string and of the archimedean Cantor string given by the firstauthor in [ – ], the notion of an archimedean fractal string was conceived anddefined by Michel Lapidus and Carl Pomerance in their investigation of the one-dimensional Weyl–Berry conjecture for fractal drums and its connection with theRiemann zeta function [ ]. The Riemann hypothesis turned out to be equivalentto the solvability of the corresponding inverse spectral problem for fractal strings, aswas established by Michel Lapidus and Helmut Maier in [ ]. The heuristic notionof complex dimensions then started to emerge and was used in a crucial way, at leastheuristically, in their spectral reformulation of the Riemann hypothesis. The precisenotion of complex dimensions, defined as the poles of a certain geometric zetafunction associated with the fractal string, was crystallized and rigorously developedby Michel Lapidus and Machiel van Frankenhuijsen in the research monograph Fractal Geometry and Number Theory [ ] and then significantly further extendedin the book Fractal Geometry, Complex Dimensions and Zeta Functions [ ].The work of Lapidus and Maier mentioned above can be summarized as follows: The inverse spectral problem for a fractal string can be solved ifand only if its dimension is not / . The inverse spectral problem for a fractal string is not solvable in dimension 1/2because the Riemann zeta function ζ ( s ) = 1 + 1 / s + 1 / s + · · · vanishes (infinitelyoften) on the critical line ℜ ( s ) = 1 /
2. Therefore, the inverse spectral problem issolvable in dimension D = 1 / ℜ ( s ) = 1 / / ] (building, in particular, on earlier work in [ ] and in [ ]), as well asMichel Lapidus and Machiel van Frankenhuijsen [ ], were led to a definition of thedual of a fractal string, interchanging the dimensions D and 1 − D . A partial answerto the question why 1 / /
2. The concept of the dual of a fractal string provides a geometricway to take the functional equation of the Riemann zeta function into considerationin the theory of complex dimensions. Such considerations would not be completeif they did not also involve the Euler product of the Riemann zeta function. Thespectral operator was introduced semi-heuristically in [ ] as the map that sendsthe geometry of a fractal string onto its spectrum. Formally, the spectral operatoradmits an (operator-valued) Euler product. In [ – ], Hafedh Herichi and Michel INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 3 Lapidus have rigorously defined and studied the spectral operator, within a properfunctional analytic setting. They have also reformulated the above criterion for theRiemann hypothesis in terms of a suitable notion of invertibility of the spectraloperator; see also [ , ] for a corresponding asymmetric criterion, expressed interms of the usual notion of invertibility of the spectral operator. Furthermore,they have shown that the (operator-valued or “quantized”) Euler product for thespectral operator also converges inside the critical strip 0 < ℜ ( s ) < , where allthe (nontrivial) complex zeros of the Riemann zeta function reside.In order to extend the framework of the theory of complex dimensions, withan aim towards applying ideas and techniques from number theory to the inversespectral problem, it is therefore natural to attempt developing a theory of p -adicfractal strings, and then globally, an ad`elic theory of fractal strings. Further layingout the foundations for such a theory is one of the main long-term goals of thispaper.We note that nonarchimedean p -adic analysis has been used in various areasof mathematics, such as arithmetic geometry, number theory and representationtheory, as well as of mathematical and theoretical physics, such as string theory,cosmology, quantum mechanics, relativity theory, quantum field theory, statisticaland condensed matter physics; see, e.g., [ ? , , , , , ] and the relevant refer-ences therein. (See also, e.g., [ – ] and [ , ].) In Number theory as the ultimatephysical theory, [ ], Igor V. Volovich has suggested that p -adic numbers can pos-sibly be used in order to describe the geometry of spacetime at very high energies(and hence, very small scales, i.e., below the Planck or the string scale) becausethe measurements in the ‘archimedean’ geometry of spacetime at fine scales do nothave any certainty. Furthermore, several authors, including Stephen W. Hawking,have also suggested that the fine scale structure of spacetime may be fractal; see,e.g., [ , , , , ]. Therefore, a geometric theory of p -adic fractal strings andtheir complex dimensions might be helpful in the quest to explore the geometryand fine scale structure of spacetime at high energies.On the other hand, in the book entitled In Search of the Riemann Zeros:Strings, fractal membranes and noncommutative spacetimes [ ], Michel Lapidushas suggested that fractal strings and their quantization, fractal membranes, maybe related to aspects of string theory and that p -adic (and possibly, ad`elic) analogsof these notions would be useful in this context in order to better understand theunderlying noncommutative spacetimes and their moduli spaces ([ , ]). Thetheory of p -adic fractal strings, once suitably ‘quantized’, may be helpful in furtherdeveloping some of these ideas and eventually providing a framework for unifyingthe real and p -adic fractal strings and membranes.In this paper, we further develop the geometric theory of p -adic (or nonar-chimedean) fractal strings, which are bounded open subsets of the p -adic Q p witha fractal “boundary”, along with the associated theory of complex dimensions and,especially, of fractal tube formulas in the nonarchimedean setting. This theory,which was first developed by Michel Lapidus and L˜u’ H`ung in [ – ], as well aslater, by those same authors and Machiel van Frankenhuijsen in [ ], extends thetheory of real (or archimedean) fractal strings and their complex dimensions ina natural way. Following [ – ], we introduce suitable geometric zeta functionsfor p -adic fractal strings whose poles play the role of complex dimensions for thestandard real fractal strings. We also discuss the explicit fractal tube formulas in MICHEL L. LAPIDUS, L˜U’ H`UNG, AND MACHIEL VAN FRANKENHUIJSEN the general case of (languid) p -adic fractal strings and in the special case of p -adicself-similar strings. Throughout this paper, these various results are illustrated inthe case of the nonarchimedean self-similar Cantor and the Fibonacci strings (in-troduced in [ , ]), as well as in the case of the p -adic Euler string (introducedin [ , ]), which (strictly speaking) is not self-similar. Some particular attentionis devoted to the 3-adic Cantor string (introduced and studied in [ ]), whose ‘met-ric’ boundary is the 3-adic Cantor set [ ], which is naturally homeomorphic to theclassic ternary Cantor set.The rest of this paper is organized as follows: In § §
3, we recall thedefinition of an arbitrary p -adic fractal string along with its associated geometriczeta function and complex dimensions, as well as the corresponding notions ofMinkowski dimension and content. Furthermore, in §
4, we discuss the important,but more technical, question of how to suitably define and calculate the volumeof the ‘inner’ ε -neighborhood of a p -adic fractal string. The definitions and proofsgiven in § p -adic) nature ofthe underlying geometry. In §
5, we then introduce a proper notion in this contextof Minkowski dimension and Minkowski content, building on the results of § p -adic fractalstring coincides with the abscissa of convergence of its geometric zeta function.In the process, we show that this common value coincides with the asymptoticgrowth rate of the geometric counting function of the fractal string, a result whichis new even in the archimedean setting, even though it may be implicit in [ ].Our proof also provides a new and unified derivation of the archimedean resultfor real fractal strings ( [ , ], [ , ]), by placing the archimedean and thenonarchimedean settings on the same footing; see § §
6, we then use ourprevious results (in § §
5) to express the volume of the inner tube of a p -adicfractal string as an infinite sum over all the underlying complex dimensions, therebyobtaining a nonarchimedean analog of the ‘fractal tube formula’ obtained for real (orarchimedean) fractal strings in [ , ]. (See, especially, [ , Ch. 8].) We illustratethis formula in § p -adic Euler string, the definition of which is givenin § ] for the important case of general p -adic self-similar strings,including the 3-adic Cantor string (Example 6.6 below) and the 2-adic Fibonaccistring. In §
7, we briefly discuss possible future research directions connected withthe theory developed in this paper and its predecessors, [ – ]. Finally, in § p -adic fractal strings of any rational dimension between0 and 1 and a possible connection between their construction and the Riemannhypothesis for the Riemann zeta function. We also discuss some constructions ofad`elic fractal strings and a geometric zeta function for the ad`elic Euler–Riemannstring.For more information about the theory of fractal strings (or sprays) and theircomplex dimensions, beside the books [ ], [ ], [ ] and [ ], we refer to [ ],[ ], [ ], [ – ], [ ], [ , ], [ – ], [ – ], [ ], [ ], [ ], as well as therelevant references therein.
2. Nonarchimedean Fractal Strings
INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 5 p -adic Numbers. Given a fixed prime number p , any nonzero rationalnumber x can be written as x = p v · a/b , for some integers a and b and a uniqueexponent v ∈ Z such that p does not divide a or b . The p -adic absolute value is thefunction | · | p : Q → [0 , ∞ ) given by | x | p = p − v and | | p = 0. It satisfies the strongtriangle inequality : for every x, y ∈ Q , | x + y | p ≤ max {| x | p , | y | p } . Relative to the p -adic absolute value, Q does not satisfy the archimedean propertybecause for every x ∈ Q , | nx | p will never exceed | x | p for any n ∈ N . The metriccompletion of Q with respect to the p -adic absolute value | · | p is the field of p -adicnumbers Q p . More concretely, every p -adic number z ∈ Q p has a unique p -adicexpansion z = a v p v + · · · + a + a p + a p + · · · , for some v ∈ Z and digits a i ∈ { , , . . . , p − } for all i ≥ v and a v = 0. Animportant subset of Q p is the unit ball, Z p = { x ∈ Q p : | x | p ≤ } , which can alsobe represented as follows: Z p = (cid:8) a + a p + a p + · · · : a i ∈ { , , . . . , p − } for all i ≥ (cid:9) . Using this p -adic expansion, one sees that(2.1) Z p = p − [ a =0 ( a + p Z p ) , where a + p Z p = { y ∈ Q p : | y − a | p ≤ /p } . Thus the p -adic ball Z p is self-similar to p scaled (by the factor 1/ p ) copies of itself. Note that Z p is compactand thus complete. Also, Q p is a locally compact group, and hence admits aunique translation invariant Haar measure µ H , normalized so that µ H ( Z p ) = 1. Inparticular, µ H ( a + p n Z p ) = p − n for every n ∈ Z . For general references on p -adicanalysis, we point out, e.g., [ , , , – ]. Remark . (a) The distance d p defined on Q p by d p ( x, y ) = | x − y | p iscalled an ultrametric, since it satisfies the counterpart of the above strong triangleinequality:(2.2) d p ( x, z ) ≤ max { d p ( x, y ) , d p ( y, z ) } for all x, y, z ∈ Q p . Consequently, every triangle in Q p is isosceles with the twolonger sides having the same length:If d p ( x, y ) > d p ( y, z ) then d p ( x, z ) = d p ( x, y ) . (2.3)It follows that the center can be chosen anywhere within the p -adic ball B . More-over, given any two balls B and B , either they are disjoint or one is entirelycontained in the other (i.e., B ⊆ B or B ⊆ B ). These special properties arecommon to all ultrametric spaces (i.e., all metric spaces for which the ultrametrictriangle inequality (2.2) holds).(b) By definition, Z p is the (closed) unit ball of ( Q p , d p ). Moreover, Z p hasthe remarkable property of being a ring (since for all x, y in Z p , by (2.2) again, | x + y | p ≤ max( | x | p , | y | p ) ≤
1, and | xy | p = | x | p | y | p ≤ − , R , is not stable under addition (althoughit is obviously stable under multiplication); see [ ]. Finally, since translations are MICHEL L. LAPIDUS, L˜U’ H`UNG, AND MACHIEL VAN FRANKENHUIJSEN homeomorphisms, every closed ball B = B ( a, r ) in Q p with center a has a radius r of the form r = p n ,(2.4) B ( a, r ) = a + p − n Z p = { x ∈ Q p : | x − a | p ≤ r } for some r ∈ p Z , the valuation group of the nonarchimedean field Q p . We leave it tothe reader to investigate the converse statement according to which every convexsubset of Q p is a metric ball (i.e., an interval); see, e.g., [ ].(c) ( p -adic intervals). In the sequel (as well as in part of the literature on p -adicanalysis, see, e.g., [ ]), the metric balls B = a + r Z p (with a ∈ Q p and r ∈ p Z ,as in (2.4) just above), are sometimes called the ‘intervals’ of Q p . Note that theyare not connected, in the usual topological sense, but that they are ‘convex’, in thefollowing sense: for each x, y ∈ B and α ∈ Z p , we have that αx + (1 − α ) y ∈ B .(Here and henceforth, it is useful to think of Z p ⊂ Q p as being the analogue of theunit interval [0 , ⊂ R , rather than of [ − , Q , is either equivalent to the standard archimedean ab-solute value on Q or to the nonarchimedean p -adic absolute value | · | p for someprime p . (Recall that two absolute values are said to be equivalent if they in-duce the same topology on Q ; this is the case if and only if one is a power of theother.) Therefore, infinitely many completions of Q (one for each prime p ) arenonarchimedean and R is the only completion of Q that is archimedean. For thisreason, one sometimes writes R = Q ∞ and refers to (the equivalence class of) theabsolute value | · | as the ‘place at infinity’, associated with the ‘prime at infinity’ orthe ‘real prime’; see [ ]. (We note that Ostrowski’s Theorem is usually expressedin terms of valuations rather than of absolute values. Accordingly, a place of Q isgenerally defined as an equivalence class of valuations on Q .) With this notation inmind, we see that the field Q ∞ is archimedean, whereas for any (finite) prime p , Q p is a nonarchimedean field. The theory of p -adic fractal strings developed in [ – ]is aimed, initially, at finding suitable definitions and obtaining results that parallelthose corresponding to the theory of real (or archimedean) fractal strings developedin [ ], for example. As we will see, however, although there are many analogiesbetween the archimedean and nonarchimedean theories of fractal strings, there arealso some notable differences between them; see, especially, [ ], along with [ ]and [ ]. p -adic Fractal Strings. Let Ω be a bounded open subset of Q p . Thenit can be decomposed into a countable union of disjoint open balls with radius p − n j centered at a j ∈ Q p , a j + p n j Z p = B ( a j , p − n j ) = { x ∈ Q p | | x − a j | p ≤ p − n j } , where n j ∈ Z and j ∈ N ∗ . (We shall often call a p -adic ball an interval . By ‘ball’here, we mean a metrically closed and hence, topologically open and closed ball.)There may be many different such decompositions since each ball can always bedecomposed into smaller disjoint balls [ ]; see Equation (2.1). However, thereis a canonical decomposition of Ω into disjoint balls with respect to a suitableequivalence relation, as we now explain. Definition . Let U be an open subset of Q p . Given x, y ∈ U, we write that x ∼ y if and only if there is a ball B ⊆ U such that x, y ∈ B . INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 7 It is clear from the definition that the relation ∼ is reflexive and symmetric. Toprove the transitivity, let x ∼ y and y ∼ z . Then there are balls B containing x, y and B containing y, z . Thus y ∈ B ∩ B ; so it follows from the ultrametricity of Q p that either B ⊆ B or B ⊆ B . In any case, x and z are contained in the sameball; so x ∼ z . Hence, the above relation ∼ is indeed an equivalence relation on theopen set U . By a standard argument (and since Q is dense in Q p ), one shows thatthere are at most countably many equivalence classes. Remark . The equivalence classes of ∼ can be thoughtof as the ‘convex components’ of U . They are an appropriate substitute in thepresent nonarchimedean context for the notion of connected components, which isnot useful in Q p since Z p (and hence, every interval) is totally disconnected. Notethat given any x ∈ U, the equivalence class (i.e., the convex component ) of x is thelargest ball containing x (or equivalently, centered at x ) and contained in U . Definition . A p -adic (or nonarchimedean ) fractal string L p is a boundedopen subset Ω of Q p .Thus it can be written, relative to the above equivalence relation, canonicallyas a disjoint union of intervals or balls: L p = ∞ [ j =1 ( a j + p n j Z p ) = ∞ [ j =1 B ( a j , p − n j ) . Here, B ( a j , p − n j ) is the largest ball centered at a j and contained in Ω. We mayassume that the lengths (i.e., Haar measure) of the intervals a j + p n j Z p are nonin-creasing, by reindexing if necessary. That is,(2.5) p − n ≥ p − n ≥ p − n ≥ · · · > . Note that, more generally, a p -adic fractal string can be defined as an opensubset Ω of Q p such that µ H (Ω) < ∞ . Definition . The geometric zeta function of a p -adic fractal string L p isdefined as(2.6) ζ L p ( s ) := ∞ X j =1 ( µ H ( a j + p n j Z p )) s = ∞ X j =1 p − n j s for all s ∈ C with ℜ ( s ) sufficiently large. Remark . The geometric zeta function ζ L p is well defined since the de-composition of L p into the disjoint intervals a j + p n j Z p is unique. Indeed, theseintervals are the equivalence classes of which the open set Ω (defining L p ) is com-posed. In other words, they are the p -adic “convex components” (rather than theconnected components) of Ω. Note that in the real (or archimedean) case, thereis no difference between the convex or connected components of Ω, and hence theabove construction would lead to the same sequence of lengths as in [ , § p -adic Euler String. The following p -adic Euler string is anew example of p -adic fractal string, which is not self-similar (in the sense of [ , ]). It is a natural p -adic counterpart of the elementary prime string , which is the local constituent of the completed harmonic string ; cf. [ , § MICHEL L. LAPIDUS, L˜U’ H`UNG, AND MACHIEL VAN FRANKENHUIJSEN
Let X = p − Z p . Then, by the ‘self-duplication’ formula (2.1), X = p − [ ξ =0 ( ξp − + Z p ) . We now keep the first subinterval Z p , and then decompose the next subintervalfurther. That is, we write p − + Z p = p − [ ξ =0 ( p − + ξ + p Z p ) . Again, iterating this process, we keep the first subinterval p − + p Z p in the abovedecomposition and decompose the next subinterval, p − + 1 + p Z p . Continuing inthis fashion, we obtain an infinite sequence of disjoint subintervals { a n + p n Z p } ∞ n =0 , where { a n } ∞ n =0 satisfies the following initial condition and recurrence relation: a = 0 and a n = a n − + p n − for all n ≥ . We call the corresponding p -adic fractal string, E p = ∞ [ n =0 ( a n + p n Z p ) , the p -adic Euler string. The geometric zeta function of the p -adic Euler string E p is ζ E p ( s ) = ∞ X n =0 ( µ H ( a n + p n Z p )) s = ∞ X n =0 p − ns = 11 − p − s , for ℜ ( s ) > . Therefore, ζ E p has a meromorphic extension to all of C given by the last ex-pression, which is the classic p - Euler factor (i.e., the local Euler factor associatedwith the prime p ):(2.7) ζ E p ( s ) = 11 − p − s , for all s ∈ C . Hence, the set of complex dimensions of E p is given by(2.8) D E p = { D + iν p | ν ∈ Z } , where D = σ = 0 and p = 2 π/ log p. Remark . The unit ball minus the origin is nota ball itself, but instead the infinite union Z p \{ } = ∞ [ n =0 p − [ k =1 kp n + p n +1 Z p , where every time, a small punctured neighborhood of 0, namely p n Z p \{ } , is sub-divided into smaller balls. This union is isomorphic to the Euler string: E p = 1 p Z p \ (cid:26) p (1 − p ) (cid:27) . INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 9 Remark . Note that ζ E p is the p -Euler factor of theRiemann zeta function; i.e., Y p< ∞ ζ E p ( s ) = Y p< ∞ − p − s = ∞ X n =1 n s = ζ ( s ) for ℜ ( s ) > . Recall that the meromorphic continuation ξ of the completed Riemann zeta functionhas the same (critical) zeros as ζ and satisfies the functional equation ξ ( s ) = ξ (1 − s ).We aim to form a certain ‘ad`elic product’ over all p -adic Euler strings (includingthe prime at infinity) so that the geometric zeta function of the resulting ad`elic Eulerstring E is the completed Riemann zeta function. Formally, the ad`elic Euler stringmay be written as E = O p ≤∞ E p and its geometric zeta function ζ E ( s ) would then coincide with the completed Rie-mann zeta function ξ (see [ ] and, e.g., [ ]): ζ E ( s ) = ξ ( s ) := π − s/ Γ( s/ Y p< ∞ − p − s . Remark . From the geometricpoint of view, the nonarchimedean Euler string E p is more natural than its archi-medean counterpart, the p -elementary prime string h p , described in [ , § E p has a very simple geometric definition. Since, byconstruction, E p and h p have the same sequence of lengths { p − n } ∞ n =0 , they havethe same geometric zeta function, namely, the p -Euler factor(2.9) ζ h p ( s ) := 11 − p − s of the Riemann zeta function ζ ( s ) , and hence, the same set of complex dimensions(2.10) D h p = (cid:26) iν π log p : ν ∈ Z (cid:27) . An ‘ad`elic version’ of the ‘harmonic string’ h , a generalized fractal string whosegeometric zeta function is ζ h ( s ) = ζ ( s ), or rather, of its completion ˜ h (so that ζ ˜ h ( s ) = ξ ( s )), is provided in [ , § , ∞ ) and the symbol ∗ denoting multiplicativeconvolution on (0 , ∞ ), we have that(2.11) h = ∗ p< ∞ h p and ˜ h = ∗ p ≤∞ h p . Furthermore, a noncommutative geometric version of this construction is pro-vided in [ ] in terms of the ‘prime fractal membrane’; see especially, [ , Chaps. 3and 4], along with [ ]. Heuristically, a ‘fractal membrane’ (as introduced in [ ])is a kind of ad`elic, noncommutative torus of infinite genus. It can also be thoughtof as a ‘quantized fractal string’; see [ , Chap. 3]. It is rigorously constructedin [ ] using Dirac-type operators, Fock spaces, Toeplitz algebras [ ], and associ-ated spectral triples (in the sense of [ ]); see also [ , § ] involving p -adic quantum mechanics.
3. The Geometric Zeta Function D WS Figure 1.
The screen S and the window W .The screen S is the graph (with the vertical and horizontal axes interchanged)of a real-valued, bounded and Lipschitz continuous function S ( t ): S := { S ( t ) + it | t ∈ R } . The window W is the part of the complex plane to the right of the screen S (seeFigure 1): W := { s ∈ C | ℜ ( s ) ≥ S ( ℑ ( s )) } . Let inf S := inf t ∈ R S ( t ) and sup S := sup t ∈ R S ( t ) , and assume that sup S ≤ σ, where σ = σ L p is the abscissa of convergence of ζ L p (to be precisely defined in (3.2) below). Definition . Let L p be a p -adic fractal string. If ζ L p has a meromorphiccontinuation to an open connected neighborhood of W ⊆ C , then(3.1) D L p ( W ) := { ω ∈ W | ω is a pole of ζ L p } is called the set of visible complex dimensions of L p . If no ambiguity may arise or if W = C , we simply write D L p = D L p ( W ) and call it the set of complex dimensions of L p .Moreover, the abscissa of convergence of ζ L p (where L p is defined in Equa-tion (2.6)) is denoted by σ = σ L p . Recall that it is defined by (see, e.g., [ ])(3.2) σ L p := inf ( α ∈ R : ∞ X j =1 p − n j α < ∞ ) . INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 11 Remark . In particular, if ζ L p is entire (which occurs only in the trivialcase when L p is given by a finite union of intervals), then σ L p = −∞ . Otherwise, σ L p ≥ L p is composed of infinitely many intervals) and we will see inTheorem 5.2 that σ L p < ∞ since σ L p ≤ D M ≤ , where D M = D M, L p is theMinkowski dimension of L p , to be introduced in §
5. Furthermore, it will followfrom Theorem 5.2 that for a nontrivial p -adic fractal string, σ L p = D M . Observe that since D L p ( W ) is defined as a subset of the poles of a meromorphicfunction, it is at most countable and forms a discrete subset of C .Finally, we note that it is well known that ζ L p is holomorphic for ℜ ( s ) > σ L p ;see, e.g., [ ]. Hence, D L p ⊆ { s ∈ C : ℜ ( s ) ≤ σ L p } . Remark . Archimedean or real fractal stringsare defined as bounded open subsets of the real line R = Q ∞ . They were initiallydefined in [ ], following an early example in [ ], and have been used extensivelyin a variety of settings; see, e.g., [ , , – , , – , , , , , – , – , , ] and the books [ , , , ]. Since an open set Ω ⊂ R is canonicallyequal to the disjoint union of finitely or countably many open and bounded intervals(namely, its connected components), say Ω = S ∞ j =1 I j , we may also describe a realfractal string by a sequence of lengths L = { l j } ∞ j =1 , where l j = µ L ( I j ) is the lengthor 1-dimensional Lebesgue measure of the interval I j , written in nonincreasingorder: l ≥ l ≥ l ≥ · · · . (A justification for this identification is provided by the formula for the volume V L ( ε ) of ε -inner tubes of Ω, as given by Equation (4.13) below.) Note that since µ L (Ω) < ∞ , l j → j → ∞ , except in the trivial case when Ω consists offinitely many intervals. Also observe that the 1-dimensional Lebesgue measure µ L is nothing but the Haar measure on R = Q ∞ , normalized so that µ L ([0 , . All of the definitions given above for p -adic fractal strings have a natural coun-terpart for real fractal strings. For instance, the geometric zeta function of L isinitially defined by(3.3) ζ L ( s ) = ∞ X j =1 ( µ L ( I j )) s = ∞ X j =1 l sj , for ℜ ( s ) > σ L , the abscissa of convergence of ζ L , and for a given screen S andassociated window W , the set D L = D L ( W ) of visible complex dimensions of L isgiven exactly as in (3.1) of Definition 3.1, except with L p and ζ L p replaced with L and ζ L , respectively. Similarly, σ L , the abscissa of convergence of ζ L is given asin (3.2), except with the lengths of L instead of those of L p . Moreover, it followsfrom [ , Thm. 1.10] that for any nontrivial real fractal string L , we have σ L = D M ,the Minkowski dimension of L (i.e., of its topological boundary ∂ Ω). This latterresult will be given a new proof in § , ] for a fulldevelopment of the theory of real fractal strings and their complex dimensions. p -adic Fractal Strings. In §
6, wewill obtain explicit tube formulas for p -adic fractal strings, with and without errorterm. (See Theorem 6.1 and Corollary 6.3.) We will then apply the tube formula without error term (the strongly languid case of Theorem 6.1) to the p -adic Eulerstring discussed in § § , Defns. 5.2 and 5.3] and recallthe definition of the screen S given in §
2, just before Definition 3.1).
Definition . A p -adic fractal string L p is said to be languid if its geometriczeta function ζ L p satisfies the following growth conditions: There exist real con-stants κ and C > { T n } n ∈ Z of real numbers such that T − n < < T n for n ≥
1, andlim n →∞ T n = ∞ , lim n →∞ T − n = −∞ , lim n →∞ T n | T − n | = 1 , such that • L1 For all n ∈ Z and all u ≥ S ( T n ) , | ζ L p ( u + iT n ) | ≤ C ( | T n | + 1) κ , • L2 For all t ∈ R , | t | ≥ , | ζ L p ( S ( t ) + it ) | ≤ C | t | κ . We say that L p is strongly languid if its geometric zeta function ζ L p satisfies thefollowing conditions, in addition to L1 with S ( t ) ≡ −∞ : There exists a sequenceof screens S m : t S m ( t ) for m ≥ , t ∈ R , with sup S m → −∞ as m → ∞ andwith a uniform Lipschitz bound sup m ≥ || S m || Lip < ∞ , such that • L2 ′ There exist constants
A, C > t ∈ R and m ≥ | ζ L p ( S m ( t ) + it ) | ≤ CA | S m ( t ) | ( | t | + 1) κ . Remark . (a) Intuitively, hypothesis L1 is a polynomial growth conditionalong horizontal lines (necessarily avoiding the poles of ζ L p ), while hypothesis L2 is a polynomial growth condition along the vertical direction of the screen.(b) Clearly, condition L2 ′ is stronger than L2 . Indeed, if L p is strongly languid,then it is also languid (for each screen S m separately).(c) Moreover, if L p is languid for some κ , then it is also languid for every largervalue of κ . The same is also true for strongly languid strings.(d) Finally, hypotheses L1 and L2 require that ζ L p has an analytic (i.e., mero-morphic) continuation to an open, connected neighborhood of ℜ ( s ) ≥ σ L p , while L2 ′ requires that ζ L p has a meromorphic continuation to all of C .
4. Volume of Thin Inner Tubes
In this section, we provide a suitable analog in the p -adic case of the ‘boundary’of a fractal string and of the associated inner tubes (or “inner ε -neighborhoods”).Moreover, we give the p -adic counterpart of the expression that yields the volumeof the inner tubes (see Theorem 4.6). This result will serve as a starting point in § Definition . Given a point a ∈ Q p and a positive real number r >
0, let B = B ( a, r ) = { x ∈ Q p | | x − a | p ≤ r } be a metrically closed ball in Q p , as above.(Recall that it follows from the ultrametricity of | · | p that B is topologically bothclosed and open (i.e., clopen) in Q p .) We call S = S ( a, r ) = { x ∈ Q p | | x − a | p = r } INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 13 the sphere of B . (In our sense, S also coincides with the ‘metric boundary’ of B ,as given in the next definition.)Let L p = S ∞ j =1 B ( a j , r j ) be a p -adic fractal string. We then define the metricboundary β L p of L p to be the disjoint union of the corresponding spheres, i.e., β L p = ∞ [ j =1 S ( a j , r j ) . Given a real number ε >
0, define the thick p -adic ‘inner ε -neighborhood’ (or‘ inner tube ’) of L p to be(4.1) N ε = N ε ( L p ) := { x ∈ L p | d p ( x, β L p ) < ε } , where d p ( x, E ) = inf {| x − y | p | y ∈ E } is the p -adic distance of x ∈ Q p to a subset E ⊆ Q p . Then the volume V L p ( ε ) of the thick inner ε -neighborhood of L p is definedto be the Haar measure of N ε , i.e., V L p ( ε ) = µ H ( N ε ) . Lemma . Let B = B ( a, r ) and S = S ( a, r ) , as in Definition 4.1. Then, forany positive number ε < r , we have (4.2) N ε ( B ) := { x ∈ B | d p ( x, S ) < ε } = S. Further, if r = p − m for some m ∈ Z , then for all ε < r, (4.3) µ H ( { x ∈ B | d p ( x, S ) < ε } ) = µ H ( S ) = (1 − p − ) p − m . Proof.
Clearly S ⊆ { x ∈ B | d p ( x, S ) < ε } since for any x ∈ S , d p ( x, S ) = 0.Next, fix ε with 0 < ε < r and let x ∈ B be such that d p ( x, S ) < ε. Then theremust exist y ∈ S such that | x − y | p < ε. But, since | y − a | p = r, we deduce fromthe fact that every “triangle” in Q p is isosceles [ , p. 6] that | x − a | p = | y − a | p and thus x ∈ S . This completes the proof of (4.2).We next establish formula (4.3). In light of Equation (4.2), it suffices to showthat(4.4) µ H ( S ) = (1 − p − ) p − m . Let S = S (0 ,
1) = { x ∈ Q p | | x | p = 1 } denote the unit sphere in Q p . Since S = S ( a, p − m ) = a + p m S , we have that µ H ( S ) = µ H ( S ) p − m . Next we note that B (0 ,
1) = [ m ≥ S (0 , p − m )is a disjoint union. Hence, by taking the Haar measure of B (0 , , we deduce that(4.5) 1 = ∞ X m =0 p − m ! µ H ( S ) = 11 − p − µ H ( S ) , from which (4.4) and hence, in light of the first part, (4.3) follows. (cid:3) Theorem . Let L p = S ∞ j =1 B ( a j , p − n j ) be a p -adic fractal string. Then, for any ε > , we have V L p ( ε ) = (1 − p − ) k X j =1 p − n j + X j>k p − n j (4.6) = ζ L p (1) − p k X j =1 p − n j , (4.7) where k = k ( ε ) is the largest integer such that p − n k ≥ ε . Proof.
In light of the definition of N ε = N ε ( L p ) given in Equation (4.1) andthe definition of k given in the theorem, we have that N ε = k [ j =1 S j ∪ [ j>k B j , where B j := B ( a j , p − n j ) and S j := S ( a j , p − n j ) for each j ≥ V L p ( ε ) = µ H ( N ε ) statedin Equations (4.6) and (4.7). (cid:3) Note that ζ L p (1) = P ∞ j =1 p − n j is the volume of L p (or rather, of the boundedopen subset Ω of Q p representing L p ): ζ L p (1) = µ H ( L p ) < ∞ . It is clearly independent of the choice of Ω representing L p , and so is V L p ( ε ) in lightof either (4.6) or (4.7). Corollary . The following limit exists in (0 , ∞ ) :(4.8) lim ε → + V L p ( ε ) = µ H ( β L p ) = (1 − p − ) ζ L p (1) . Proof.
This follows by letting ε → + in either (4.6) and (4.7) and notingthat k = k ( ε ) → ∞ . (cid:3) Corollary 4.4, combined with the fact that β L p ⊂ N ε ( L p ) for any ε > , naturally leads us to introduce the following definition. Definition . Given ε > , the thin p -adic ‘inner ε -neighborhood’ (or ‘innertube’ ) of L p is given by(4.9) N ε = N ε ( L p ) := N ε ( L p ) \ β L p . Then, in light of Corollary 4.4, the volume V L p ( ε ) of the thin inner ε -neighbor-hood of L p is defined to be the Haar measure of N ε and is given by(4.10) V L p ( ε ) := µ H ( N ε ) = V L p ( ε ) − µ H ( β L p ) . Note that, by construction, we now have lim ε → + V L p ( ε ) = 0 . We next state the counterpart (for thin inner tubes) of Theorem 4.3, whichis the key result that will enable us to obtain an appropriate p -adic analog of thefractal tube formula (in §
6) as well as of the notions of Minkowski dimension andcontent (in § Theorem . Let L p = S ∞ j =1 B ( a j , p − n j ) be a p -adic fractal string. Then, for any ε > , we have V L p ( ε ) = p − X j>k p − n j = p − X j : p − nj <ε p − n j (4.11) = p − (cid:18) ζ L p (1) − k X j =1 p − n j (cid:19) , (4.12) where k = k ( ε ) is the largest integer such that p − n k ≥ ε , as before. INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 15 Proof.
In view of Theorem 4.3 and Corollary 4.4, the result follows immedi-ately from Equation (4.10) in Definition 4.5. (cid:3)
Remark . Observe that because the center a of a p -adic ball B = B ( a, p − n )can be chosen arbitrarily without changing its radius p − n , the metric boundary ofa ball, βB = S = S ( a, p − n ) , depends on the choice of a . Note, however, thatin view of Equation (4.3) in Lemma 4.2, its volume µ H ( S ) depends only on theradius of B . Similarly, even though the decomposition of a p -adic fractal string Ω(i.e., L p ) into maximal balls B j = B j ( a j , p − n j ) is canonical, ‘the’ metric boundaryof L p , β L p = S ∞ j =1 S ( a j , r j ) , depends on the choice of the centers a j . However,according to Corollary 4.4, µ H ( β L p ) is independent of this choice and hence, neither V L p ( ε ) = µ H ( N ε ( L p )) nor V L p ( ε ) = µ H ( N ε ( L p )) depends on the choice of thecenters. Indeed, in light of Theorem 4.3 and Theorem 4.6, V L p ( ε ) and V L p ( ε )depend only on the choice of the p -adic lengths p − n j , and hence solely on the p -adic fractal string L p , viewed as a nonincreasing sequence of positive numbers, andnot on the geometric representation Ω of L p , let alone on the choice of the centersof the balls of which Ω is composed.Although it is not entirely analogous to it, this situation is somewhat reminis-cent of the fact that the volume V L ( ε ) of the inner ε -neighborhoods of an archime-dean fractal string depends only on its lengths { l j } ∞ j =1 and not on the representativeΩ of L as a bounded open set; see Equation (4.13) and the discussion surroundingit in Remark 4.8. Remark . Recall that V L p ( ε ) does not tend to zero as ε → + , but that instead it tendsto the positive number (1 − p − ) ζ L p (1) , whereas V L p ( ε ) does tend to zero. This isthe reason why the Minkowski dimension must be defined in terms of V L p ( ε ) (as willbe done in §
5) rather than in terms of V L p ( ε ) . Indeed, if V L p ( ε ) were used instead,then every p -adic fractal string would have Minkowski dimension 1. This would bethe case even for a trivial p -adic fractal string composed of a single interval, forexample. This is also why, in the p -adic case, we will focus only on the tube formulafor V L p ( ε ) rather than for V L p ( ε ), although the latter could be obtained by meansof the same techniques.Note the difference between the expressions for V L ( ε ) in the case of an archi-medean fractal string L and for its nonarchimedean thin (resp., thick) counterpart V L p ( ε ) (resp., V L p ( ε )) in the case of a p -adic fractal string L p . Compare Equation(8.1) of [ ] (which was first obtained in [ ]),(4.13) V L ( ε ) = X j : l j ≥ ε ε + X j : l j < ε l j , with Equations (4.11)–(4.12) in Theorem 4.6. (Here, we are using the notationof Remark 3.3, to which the reader is referred to for a brief introduction to realfractal strings.) It follows, in particular, that V L ( ε ) is a continuous function of ε on (0 , ∞ ), whereas V L p ( ε ) (and hence also V L p ( ε )) is discontinuous (because it isa step function with jump discontinuities at each point p − n j , for j = 1 , , , . . . ).The above discrepancies between the archimedean and the nonarchimedean caseshelp explain why the tube formula for real and p -adic fractal strings have a similarform, but with different expressions for the corresponding ‘tubular zeta function’(in the sense of [ , ]). We note that a minor aspect of these discrepancies is that ε is now replaced by ε. Interestingly, this is due to the fact that the unit interval[0 ,
1] has inradius 1/2 in R = Q ∞ whereas Z p has inradius 1 in Q p . Recall that the inradius of a subset E of a metric space is the supremum of the radii of the ballsentirely contained in E .Finally, we note that for an archimedean fractal string L , there is no reasonto distinguish between the ‘thin volume’ V L and the ‘thick volume’ V L , as wenow explain. Indeed, the archimedean analogue β L of the metric boundary is acountable set, and hence has measure zero, no matter which geometric realizationΩ one chooses for L . More specifically, in the notation of Remark 3.3, β L consistsof all the endpoints of the open intervals I j (the connected components of Ω, orequivalently, its convex components). Hence, µ L ( β L ) = 0 and so V L ( ε ) := V L ( ε ) − µ L ( β L ) = V L ( ε ) , as claimed.For example, if L is the ternary Cantor string CS , then β L is the countableset consisting of all the endpoints of the ‘deleted intervals’ in the construction ofthe real Cantor set C . In other words, β L is the set T of ternary points (whichhas measure zero because it is countable). Hence, the metric boundary β L of CS is dense in ∂ L , the topological boundary of CS , and which in the present case,coincides with the ternary Cantor set C . Also note that the fact that C = ∂ L (andnot T = β L ) has measure zero is purely coincidental and completely irrelevant here.Indeed, the same type of argument would apply if L were any archimedean fractalstring, even if µ L ( ∂ L ) > ∂ L is a ‘fat Cantor set’(i.e., a Cantor set of positive measure) or, more generally, if ∂ L is a ‘fat fractal’ (inthe sense of [ , ]). The underlying reason is that in the archimedean case, thetopological boundary ∂ L = ∂ Ω is disjoint from Ω (since Ω is open), and hence, doesnot play any role in the computation of V L ( ε ) or of V L ( ε ). By contrast, it is nottrue that the metric boundary β L and the geometric representation Ω are disjoint(since, in fact, β L ⊆
Ω), but what is remarkable is that the Minkowski dimensionof β L coincides with that of its closure, and hence (in most cases of interest), with D M, L . As a first application of Theorem 4.6,we can obtain, via a direct computation, a tube formula for the p -adic Euler string E p ; that is, an explicit formula for the volume of the thin inner ε -neighborhood, V E p ( ε ), as given in Definition 4.5.Let E p be the p -adic Euler string defined in § ε >
0, let k be thelargest integer such that µ H ( a k + p k Z p ) = p − k ≥ ε ; then k = [log p ε − ]. (Here, for x ∈ R , we write x = [ x ] + { x } , where [ x ] is the integer part and { x } is the fractionalpart of x ; i.e., [ x ] ∈ Z and 0 ≤ { x } < V E p ( ε ) = p − ∞ X n = k +1 p − n = p − p − p − k = p − p − p − log p ε − (cid:18) p (cid:19) −{ log p ε − } , since k = log p ε − − { log p ε − } . Next, the Fourier series expansion for b −{ x } isgiven by (see [ , Eq. (1.13)])(4.14) b −{ x } = b − b X n ∈ Z e πinx log b + 2 πin , INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 17 Applying it with b = 1 /p and x = log p ε − , we find V E p ( ε ) = p − p − p − p X n ∈ Z ε − in p − in p = 1 p log p X ω ∈D E p ε − ω − ω . (4.15)Finally, in the last equality, we have used Equation (2.8) for the set of complexdimensions D E p of E p .
5. Minkowski Dimension
In the sequel, the (inner) Minkowski dimension and the (inner) Minkowskicontent of a p -adic fractal string L p (or, equivalently, of its metric boundary β L p ,see Definition 4.1) is defined exactly as the corresponding notion for a real fractalstring (see [ , Defn. 1.2]), except for the fact that we now use the definition of V ( ε ) = V L p ( ε ) provided in Equation (4.10) of Definition 4.5. (For reasons that willbe clear to the reader later on in this section, we denote by D M = D M, L p insteadof by D = D L p the Minkowski dimension of L p . ) More specifically, the Minkowskidimension of L p is given by(5.1) D M = D M, L p := inf (cid:8) α ≥ | V L p ( ε ) = O ( ε − α ) as ε → + (cid:9) . Furthermore, L p is said to be Minkowski measurable , with
Minkowski content M ,if the limit(5.2) M = lim ε → + V L p ( ε ) ε − (1 − D M ) exists in (0 , ∞ ) . Remark . Note that since V L p ( ε ) = V L p ( ε ) − µ H ( β L p ), the above definitionof the Minkowski dimension is somewhat analogous to that of “exterior dimension”,which is sometimes used in the archimedean case to measure the roughness of a‘fat fractal’ (i.e., a fractal with positive Lebesgue measure). The notion of exteriordimension has been useful in the study of aspects of chaotic nonlinear dynamics;see, e.g., [ ] and the survey article [ ].The goal of the rest of this section is to establish the following theorem, whichis the exact analogue for p -adic fractal strings of [ , Thm. 1.10], which was firstobserved in [ , ] by using a result of [ ]. (Recall that σ L p is defined in Equation(3.2) of §
3. Also note that we need to assume that L p has infinitely many lengthssince if L p is composed of finitely many intervals, then σ L p = −∞ and D M = D = 0;see formula (5.7) below for the definition of D .) Theorem . Let L p be a p -adic fractal string composed of infinitely manyintervals. Then the Minkowski dimension D M = D M, L p of L p equals the abscissaof convergence σ L p of the geometric zeta function ζ L p . That is, D M = σ L p . Theorem 5.2 will be established in Theorem 5.4 below in greater generality,namely for any summable sequence of positive numbers l j . This is the object of thetechnical Lemma 5.3, which is of independent interest.When applied to a p -adic fractal string, the following lemma relates the thinvolume with the zeta function. For completeness, but independently of this, we also formulate the counterpart for the counting function of the reciprocal lengths ofan arbitrary fractal string. The lemma holds in general, independently of the factthat in the present situation, the lengths are powers of p . Recall from (4.11) that V ( ε ) = 1 p X j : l j ≤ ε l j , writing V instead of V L p since what follows holds for arbitrary infinite sequences ofpositive numbers l j such that P ∞ j =1 l j is convergent. Also, the geometric countingfunction of L := { l j } ∞ j =1 , N ( x ) := X l j ≥ /x , (5.3)is the number of reciprocal lengths up to x >
0, and ζ L ( s ) := ∞ X j =1 l sj , (5.4)at least for all s ∈ C such that R s > Lemma . We have the following two expressions for ζ L ( s ) : ζ L ( s ) = ζ L (1) l s − + (1 − s ) Z l pV ( ε ) ε s − dε, (5.5) and ζ L ( s ) = s Z ∞ N ( x ) x − s − dx. (5.6) Both expressions converge exactly when P ∞ j =1 l sj converges. Proof.
For n >
0, we compute(1 − s ) Z l l n pV ( ε ) ε s − dε = n − X j =1 (1 − s ) Z l j l j +1 pV ( ε ) ε s − dε = n − X j =1 ∞ X k = j +1 l k (1 − s ) Z l j l j +1 ε s − dε, since for l j +1 ≤ ε < l j , the function pV ( ε ) is constant, equal to P k>j l k . Wecompute the integral to obtain(1 − s ) Z l l n pV ( ε ) ε s − dε = n − X j =1 ∞ X k = j +1 l k (cid:0) l s − j +1 − l s − j (cid:1) . Next, we split the sum and interchange the order of summation, to obtain n X j =2 ∞ X k = j l k l s − j − n − X j =1 ∞ X k = j +1 l k l s − j = ∞ X k =2 l k min { k,n } X j =2 l s − j − ∞ X k =2 l k min { k,n }− X j =1 l s − j . INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 19 In this formula, the two double sums clearly converge, since P k ≥ l k converges.Simplifying, we obtain(1 − s ) Z l l n pV ( ε ) ε s − dε = ∞ X k =2 l k l s − { k,n } − ∞ X k =2 l k l s − = n − X k =2 l sk + ∞ X k = n l k l s − n − l s − ∞ X k =2 l k . Now, l k l s − n ≤ l sk for k ≥ n (if s ≥
1, we estimate instead l s − n ≤
1, provided n is so large that l n ≤ n approach infinity if and only if P ∞ k =2 l sk converges, and then the middle sum converges to zero. In that case, we obtain P k ≥ l sk − l s − ζ L (1) = ζ L ( s ) − ζ L (1) l s − for the limit.In a similar way, we compute s Z l − n N ( x ) x − s − dx = n − X j =1 s Z l − j +1 l − j N ( x ) x − s − dx = n − X j =0 j (cid:0) l sj − l sj +1 (cid:1) , since N ( x ) = 0 for x < l − , and N ( x ) = j for l − j ≤ x < l − j +1 . We find s Z l − n N ( x ) x − s − dx = n − X j =1 jl sj − n X j =1 ( j − l sj = n X j =1 l sj − nl sn . Now, nl sn ≤ P nj =[ n/ l sj , provided s ≥
0, so we can let n approach infinity if andonly if P ∞ j =1 l sj converges, in which case we find the value ζ L ( s ) for the limit, againsince the tail P ∞ j =[ n/ l sj converges to zero. (cid:3) Recall that the Minkowski dimension D M was defined in (5.1) above. We alsodefine the growth rate of L (or asymptotic growth rate of the geometric countingfunction N := N L ) by D := inf (cid:8) α ≥ | N ( x ) = O ( x α ) as x → ∞ (cid:9) . (5.7) Theorem . Assume that the hypotheses of Theorem 5.2 are satisfied. Then σ L , the abscissa of convergence of ζ L , coincides with D M and with D . That is, D M = σ L = D . Proof.
Let α > D M . Since, by definition of D M , V ( ε ) ≤ Aε − α , then(1 − s ) Z l V ( ε ) ε s − dε ≤ A (1 − s ) Z l ε s − α − dε. (Here, A is some suitable positive constant.) This integral converges for all realnumbers s > α ; hence, by the foregoing lemma (Lemma 5.3), σ ≤ α , where (fornotational simplicity) σ = σ L denotes the abscissa of convergence of L := { ℓ j } ∞ j =1 .Since this holds for all α ∈ R such that α > D M , we conclude that σ ≤ D M .Conversely, if α < D M , then V ( ε ) is not O ( ε − α ) as ε → + . This means thatthere exists a sequence { ε n } ∞ n =0 converging to 0 , with l ≥ ε > ε > ε > . . . andsuch that V ( ε j ) ≥ ε − αj for every j ≥
1. Moreover, we may choose the sequence tobe exponentially decreasing; say, ε n +1 < ε n / n ≥
1. Then, for s ≤ − s ) Z l V ( ε ) ε s − dε ≥ ∞ X j =1 (1 − s ) Z ε j − ε j ε − αj ε s − dε, since V ( ε ) is increasing. We then estimate(1 − s ) Z ε j − ε j ε s − dε = ε s − j − ε s − j − ≥ ε s − j (cid:0) − s − (cid:1) , to obtain (1 − s ) Z l V ( ε ) ε s − dε ≥ ∞ X j =1 ε s − αj (cid:0) − s − (cid:1) . For all s ∈ R such that s ≤ α , this sum diverges. Again by Lemma 5.3, we concludethat σ ≥ α . This holds for all α < D M ; hence, σ ≥ D M . Together with the firstpart, we conclude that σ = D M .Next, we show that σ = D . If α > D , then N ( x ) ≤ Ax α for some A >
0. Thenfor all s ∈ R such that s > α , and noting that N ( x ) vanishes for x < l − , we havethat s Z ∞ l − N ( x ) x − s − dx ≤ As − α l s − α . Hence, according to Lemma 5.3, σ ≤ α . Since this holds for all α > D , we concludethat σ ≤ D .Conversely, if α < D , then N ( x ) is not O ( x α ). This means that there existsan unbounded sequence { x j } ∞ j =0 tending to ∞ , with l − ≤ x < x < x < . . . and such that N ( x j ) ≥ x αj for every j . Moreover, we choose the sequence to beexponentially increasing, x n +1 > x n . Then, for s ≥ s Z ∞ l − N ( x ) x − s − dx ≥ ∞ X j =0 s Z x j +1 x j x αj x − s − dx, since N ( x ) is increasing. We estimate s R x j +1 x j x − s − dx ≥ x − sj (cid:0) − − s (cid:1) , to obtain s Z ∞ l − N ( x ) x − s − dx ≥ ∞ X j =1 x α − sj (cid:0) − − s (cid:1) . For s ≤ α , this sum diverges. We conclude that σ ≥ α . This holds for all α < D ,hence σ ≥ D , and it follows that σ = D .Combining all of the above steps, we conclude that σ = D M = D, as desired. (cid:3) Corollary . For any p -adic fractal string L p with infinitely many lengths,we have ≤ D M = σ L p ≤ . Furthermore, we have that D M = σ L p = D, where D is the growth rate of L p defined by (5.7) . For (ordinary) archimedean fractal strings, the Min-kowski dimension also determines the abscissa of convergence of the geometric zetafunction, in an analogous manner. The advantage of our new proof is that it yieldsa unified approach to both the archimedean and nonarchimedean (or p -adic) cases.It also establishes in the process the new result according to which D M is notonly equal to σ L (the abscissa of convergence of L ) but also to D (the asymp-totic growth rate of L ), a useful fact which was only implicit in earlier work (suchas [ , , , , ]), even for real (or archimedean) fractal strings and is explicitlyneeded, for example, in [ – ].We provide here the details of our unified approach, but by focusing, of course,on the real (or archimedean) case. INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 21 The geometric zeta function of a real fractal string L = { ℓ j } ∞ j =1 , with l ≥ l ≥ l ≥ · · · →
0, is given by (5.4), just as in the p -adic case, with an abscissa ofconvergence defined by(5.8) σ L = inf ( α ∈ R : ∞ X j =1 l αj < ∞ ) , entirely analogous to (3.2). We assume an infinite number of positive lengths witha finite total length, ζ L (1) = ∞ X j =1 l j < ∞ ;so that 0 ≤ σ L ≤ N L (and hence D ) are defined in thesame way, it immediately follows that (5.6) holds in the archimedean case as well,and consequently, σ L = D .On the other hand, the formula for the volume of the tubular neighborhoodsis different, due to the different geometry of the boundary of balls in Q p (as wehave seen in §
4) and of intervals in R . In particular, the real unit interval [0 ,
1] hasinradius 1/2 in R = Q ∞ whereas the p -adic unit ball Z p has inradius 1 in Q p (seealso Remark 4.8). For real fractal strings, V L ( ε ) is given by (4.13), and hence theformula corresponding to (5.5) is ζ L ( s ) = sζ L (1) l s − + 2 s (1 − s ) Z l / V L ( ε )(2 ε ) s − dε, (5.9)valid for ℜ s > D M , where D M is defined by (5.1). This can be proved by a methodsimilar to the proof of Lemma 5.3, but we give here an alternative proof. Thefunction V L = V L ( ε ) is continuous and piecewise differentiable for ε >
0, withderivative V ′L ( ε ) = 2 N L (cid:16) ε (cid:17) . Integrating by parts, we obtain2 s (1 − s ) Z l / V L ( ε )(2 ε ) s − dε = − s (cid:2) V L ( ε )(2 ε ) s − (cid:3) l / + 2 s Z l / N L (1 / ε )(2 ε ) s − dε = − sζ L (1) l s − + s Z ∞ l − N L ( x ) x − − s dx, which by (5.6) equals − sζ L (1) l s − + ζ L ( s ) since N L ( x ) = 0 for x < l − . Remark . As an alternative, the zeta function is also given by ζ L ( s ) = 2 s (1 − s ) Z ∞ V L ( ε )(2 ε ) s − dε. This expression only converges for D M < ℜ s < ℜ s > D M .In addition, it follows from the proof that the formula converges if and only if theseries for ζ L converges. This implies that σ L ≤ D M . In order to prove the converse inequality, we construct, just as in the proof of Theorem 5.4, a sequence of positive ε -values decreasing exponentially fast to zero in order to show that (5.9) does notconverge for ℜ s < D M . It follows that σ L = D M .We conclude that for a real (or archimedean) fractal string, we have D M = σ L = D , just as was shown in the first part of this section for a p -adic (or nonarchimedean)fractal string, and thereby completing the statement and the proof of Theorem 5.2and Theorem 5.4 (now extended to the real case), as well as providing a unifiedtreatment of both the archimedean and nonarchimedean cases.
6. Explicit Tube Formulas for p -adic Fractal Strings The following result is the counterpart in this context of Theorem 8.1 of [ ],the distributional tube formula for real fractal strings. It is established by using, inparticular, the extended distributional explicit formula of [ , Thms. 5.26 and 5.27],along with the expression for the volume of thin inner ε -tubes obtained in Theorem4.6. We now state our general nonarchimedean (or p -adic) fractal tube formula inthis context. Theorem p -adic explicit tube formula) . ( i ) Let L p be a languid p -adicfractal string ( as in the first part of Definition 3.4 ) , for some real exponent κ anda screen S that lies strictly to the left of the vertical line ℜ ( s ) = 1 . Further assumethat σ L p < . ( Recall from Corollary 5.5 that we always have σ L p ≤ . Moreover, if L p is self-similar, then σ L p < . ) Then the volume of the thin inner ε -neighborhoodof L p is given by the following distributional explicit formula, on test functions in D (0 , ∞ ) , the space of C ∞ functions with compact support in (0 , ∞ ) :(6.1) V L p ( ε ) = X ω ∈D L p ( W ) res (cid:18) p − ζ L p ( s ) ε − s − s ; ω (cid:19) + R p ( ε ) , where D L p ( W ) is the set of visible complex dimensions of L p ( as given in Definition3.1 ) . Here, the distributional error term is given by (6.2) R p ( ε ) = 12 πi Z S p − ζ L p ( s ) ε − s − s ds and is estimated distributionally ( in the sense of [ , Defn. 5.29]) by (6.3) R p ( ε ) = O ( ε − sup S ) , as ε → + . ( ii ) Moreover, if L p is strongly languid ( as in the second part of Definition 3.4 ) ,then we can take W = C and R p ( ε ) ≡ , provided we apply this formula to testfunctions supported on compact subsets of [0 , A ) . The resulting explicit formulawithout error term is often called an exact tube formula in this case. Proof.
Since the proof of Theorem 6.1 parallels that of its counterpart forreal fractal strings (see [ , Thm. 8.7]), we only provide here the main steps. Wewill explain, in particular, why the p -adic tube formula takes a different form thanin the real case. As will be clear from the proof, it all goes back to the differencebetween Theorem 4.6 and its archimedean analogue (see Equation (4.13) aboveor [ , Equation (8.1)]). INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 23 According to Theorem 4.6, V L p ( ε ) = 1 p X j : p − nj <ε p − n j = 1 p Z ∞ /ε x η ( dx ) = hP [0] η , v ε i , where P [0] η = η := P ∞ j =1 δ { p nj } is viewed as a distribution and v ε ( x ) := ( x ≤ /ε / ( px ) if x > /ε. Fix ϕ ∈ D (0 , ∞ ). Then Z ∞ ϕ ( ε ) v ε ( x ) dε = 1 px Z ∞ /x ϕ ( ε ) dε = p − ϕ ( x ) , where ϕ is a smooth, but not compactly supported, test function, given by ϕ ( x ) := 1 x Z ∞ /x ϕ ( ε ) dε. Thus h V L p ( ε ) , ϕ i = Z ∞ ϕ ( ε ) Z ∞ v ε ( x ) η ( dx ) dε = D P [0] η , p − ϕ ( x ) E . (6.4)The Mellin transform of ϕ is computed to be˜ ϕ ( s ) = 11 − s ˜ ϕ (2 − s ) for ℜ s < . (6.5)Furthermore, by analytic continuation, and since ˜ ϕ ( s ) is entire for ϕ ∈ D (0 , ∞ ),the equality in (6.5) continues to hold for all s ∈ C . Now, let Ψ = p − ϕ . Its Mellin transform is e Ψ( s ) = p − − s ˜ ϕ (2 − s ) , which holds for all s ∈ C . Note that it follows from our previous discussion that e Ψ( s ) is meromorphic in all of C , with a single, simple pole at s = 1.Next, we deduce from (6.4) and [ , Thm. 5.26] (the extended distributionalexplicit formula) that h V L p ( ε ) , ϕ i = X ω ∈D L p res (cid:16) ζ L p ( s ) e Ψ( s ); ω (cid:17) + R p ( ε )= Z ∞ X ω ∈D L p res (cid:18) ζ L p ( s ) ε − s p (1 − s ) ; ω (cid:19) ϕ ( ε ) dε + Z ∞ R p ( ε ) ϕ ( ε ) dε. Therefore, V L p ( ε ) = X ω ∈D L p ( W ) res (cid:18) ζ L p ( s ) ε − s p (1 − s ) ; ω (cid:19) + R p ( ε ) , where the distribution R p ( ε ) is given by (6.2) and is estimated distributionally asin (6.3). In closing this proof, we note that in the strongly languid case, we use [ ,Thm. 5.27] in order to conclude that (6.1) holds with R p ( ε ) ≡ . (cid:3) Remark . We may rewrite the (typically infinite) sum in (6.1) as follows:(6.6) X ω ∈D L p ( W ) res( ζ L p ( ε ; s ); s = ω ) , where (by analogy with the definitions and results in [ , ]),(6.7) ζ L p ( ε ; s ) := p − ζ L p ( s ) ε − s − s is called the nonarchimedean tubular zeta function of the p -adic fractal string L p .By contrast, the archimedean tubular zeta function (in the present one-dimen-sional situation) of a real fractal string L is given by(6.8) ζ L ( ε ; s ) := ζ L ( s )(2 ε ) − s s (1 − s ) , and the analog of the above sum in the archimedean tube formula of [ ] (asrewritten in [ ]) is given as in (6.6), except with L p replaced by L and with D L ( W ) ∪ { } instead of D L p ( W ). Note that ζ L ( ε ; s ) typically has a pole at s = 0,whereas ζ L p ( ε ; s ) does not. Corollary p -adic fractal tube formula) . If, in addition to the hypothesesin Theorem 6.1, we assume that all the visible complex dimensions of L p are simple,then (6.9) V L p ( ε ) = X ω ∈D L p ( W ) c ω ε − ω − ω + R p ( ε ) , where c ω = p − res (cid:0) ζ L p ; ω (cid:1) . Here, the error term R p is given by (6.2) and isestimated by (6.3) in the languid case. Furthermore, we have R p ( ε ) ≡ in thestrongly languid case, provided we choose W = C . Remark . In [ , Ch. 8], under different sets of assumptions, both distri-butional and pointwise tube formulas are obtained for archimedean fractal strings(and also, for archimedean self-similar fractal strings). (See, in particular, Theo-rems 8.1 and 8.7, along with § ].) At least for now, in the nonarchimedeancase, we limit ourselves to discussing distributional explicit tube formulas. We ex-pect, however, that under appropriate hypotheses, one should be able to obtain apointwise fractal tube formula for p -adic fractal strings and especially, for p -adicself-similar strings. In fact, for the simple examples of the nonarchimedean Can-tor, Euler and Fibonacci strings, the direct derivation of the fractal tube formula(6.9) yields a formula that is valid pointwise and not just distributionally. (See,in particular, § Example p -adic Euler string) . We now ex-plain how to recover from Theorem 6.1 (or Corollary 6.3) the tube formula for theEuler string E p obtained via a direct computation in § W = C ) that(6.10) V E p ( ε ) = 1 p X ω ∈D E p res( ζ E p ; ω ) ε − ω − ω , INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 25 which is exactly the expression obtained for V E p ( ε ) in formula (4.15) of § ζ E p ; ω ) = 1log p for all ω ∈ D E p . (This latter observation follows easily from the expression of ζ L p obtained in Equation (2.7).) Note that Corollary 6.3 can be applied here in thestrongly languid case when W = C and R p ( ε ) ≡ § E p are simple and ζ E p is clearly strongly languidof order κ := 0 and with the constant A := p − . Furthermore, formula (6.10) canbe rewritten in the following more concrete form:(6.11) V E p ( ε ) = 1 p log p X n ∈ Z ε − in p − in p , since D E p = { in p : n ∈ Z } and p = 2 π/ log p (as in Equation (2.8) of § X n ∈ Z ε − in p − in p converges pointwise because the associated Fourier series P n ∈ Z e πinx − in p is pointwiseconvergent on R , it follows that the p -adic fractal tube formulas (6.10)–(6.11) ac-tually converge pointwise rather than just distributionally. Example . Inthis example, we explain how to derive the exact fractal tube formula for CS , the3-adic Cantor string introduced in [ ] and further studied in [ – ].By construction, the complement of CS in Z is the 3-adic Cantor set C ,which is a nonarchimedean self-similar set (as introduced in [ , ]); so that C isthe unique nonempty compact subset K of Z (or of Q ) which is the solution ofthe fixed point equation(6.12) K = ϕ ( K ) ∪ ϕ ( K ) , for some suitable affine similarity transformations ϕ , ϕ from Z to itself; morespecifically, we have that ϕ ( x ) = 3 x and ϕ ( x ) = 2 + 3 x , for all x ∈ Z . We referto [ – ] for more information concerning the properties of C and CS as wellas for corresponding figures. (See [ ] for the general definition of self-similar setsin complete metric spaces, and [ ] for a detailed discussion in the usual case ofEuclidean spaces.)Let ε >
0. We have that ζ CS ( s ) = 3 − s − · − s , for all s ∈ C and hence D CS = { D + iν p | ν ∈ Z } , will all the complex dimensions being simple and where D := log p :=2 π/ log 3. Furthermore, we have thatres( ζ CS ; ω ) = 12 log 3 , independently of ω ∈ D CS , and so the exact fractal tube formula for the nonar-chimedean Cantor string is found to be(6.13) V CS ( ε ) = 13 X ω ∈D CS res( ζ CS ; ω ) ε − ω − ω . Note that since CS has simple complex dimensions, we may also apply Corol-lary 6.3 (in the strongly languid case when W = C ) in order to precisely recoverEquation (6.13).We may rewrite (6.13) in the following form: V CS ( ε ) = ε − D G CS (log ε − ) , where G CS is the nonconstant periodic function (of period 1) on R given by G CS ( x ) := 16 log 3 X n ∈ Z e πinx − D − in p . Finally, we note that since the Fourier series X n ∈ Z e πinx − D − in p is pointwise convergent on R , the above direct computation of V CS ( ε ) showsthat (6.13) actually holds pointwise rather than just distributionally.In closing this example, we note that we could similarly use Theorem 6.1 (orCorollary 6.3) to obtain an exact fractal tube formula for the 5-adic Cantor stringrecently introduced in [ ] and defined in a way analogous to the 3-adic Cantorstring from [ ]. Remark . The 3-adic Cantor string discussed in Example 6.6 is an exam-ple of a p -adic (here, 3-adic) self-similar string. Another example of a p -adic (ornonarchimedean) self-similar string is the 2-adic Fibonacci string, whose complexdimensions are distributed periodically along two vertical lines (instead of a singleone as in the case of a 3-adic Cantor string). (See [ , ]; furthermore, see [ ]for the corresponding exact pointwise tube formula.) In general, a (nontrivial) p -adic self-similar string L p is always lattice (that is, its scaling ratios are all integerpowers of a single number, necessarily p ; see [ , ]. Therefore, unlike for real(or archimedean) fractal strings (compare with [ , Chs. 2 & 3]), which can beeither lattice or nonlattice, the complex dimensions of L p are always periodicallydistributed along finitely many vertical lines, the right most of which is the verticalline { R s = D M } , where D M is the Minkowski dimension of L p . The correspondingfractal tube formulas, illustrating our main theorem in this section (Theorem 6.1)in order to obtain fractal tube formulas for general p -adic self-similar fractal strings,are provided in [ ].In order to avoid unnecessary repetitions, we refer the interested reader to [ ](and [ ]) for those special but important examples of fractal tube formulas. Weonly mention the following two interesting facts:(i) Because on each relevant vertical line, the complex dimensions form anarithmetic progression (with a progression or period independent of theline) and have the same multiplicities, the corresponding term in the as-sociated fractal tube formula can be written as a suitable power functiontimes a periodic function (of x := log( ε − )). (This is so assuming that INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 27 the complex dimensions on that line are simple, which is always the case,for instance, of the right most vertical line { R s = D M } .)(ii) In all of the concrete examples of p -adic self-similar strings studied in[ , ], including the 3-adic Cantor string and the 2-adic Fibonacci string,the corresponding exact fractal tube formula can be shown to converge pointwise (rather than distributionally, as in Theorem 6.1). We conjecturethat at least in the case of simple complex dimensions, the exact fractaltube formula of a p -adic self-similar string always converges pointwise (andnot just distributionally, as in Theorem 6.1). (Such a result is establishedin [ , § p -adic (not necessarily self-similar) fractal strings, with or without anerror term, which (under suitable hypotheses) would be valid pointwise.We note that in the archimedean case (i.e., for real fractal strings) such apointwise fractal tube formula is available under rather general conditions;see [ , §
7. Possible Extensions
We close this paper by providing possible directions for future investigations inthis area. In Remark 6.7, we have already mentioned the problem of obtaining apointwise fractal tube formula, analogous to our distributional fractal tube formula(Theorem 6.1) in the nonarchimedean case and to the pointwise tube formula ob-tained in the archimedean case in [ , § , § It would be interestingto unify the archimedean and nonarchimedean settings by appropriately defining ad`elic fractal strings, and then studying the associated spectral zeta functions (asis done for standard archimedean fractal strings in [ , ] and [ , , , , ]).To this aim, the spectrum of these ad`elic fractal strings should be suitably definedand its study may benefit from Dragovich’s work [ ] on ad`elic quantum harmonicoscillators. In the process of defining these ad`elic fractal strings, we expect tomake contact with the notion of a fractal membrane (or “quantized fractal string”)introduced in [ , Ch. 3] and rigorously constructed in [ ] as a Connes-type non-commutative geometric space [ ]; see also [ , § ].(See also Remark 2.9 above.) We note that a geometric construction of certainad`elic fractal strings is proposed in the epilogue ( §
8) below.
As was shownin [ ] and recalled in Remark 6.7, there can only exist lattice p -adic self-similarstrings, because of the discreteness of the valuation group of Q p . However, in the archimedean setting, there are both lattice and nonlattice self-similar strings;see [ , Chs. 2 & 3]. We expect that by suitably extending the notion of p -adicself-similar string to Berkovich’s p -adic analytic space [ , ], it can be shown that p -adic self-similar strings are generically nonlattice in this broader setting. Further-more, we conjecture that every nonlattice string in the Berkovich projective linecan be approximated by lattice strings with increasingly large oscillatory periods(much as occurs in the archimedean case [ , Ch. 3]). Finally, we expect that, bycontrast with what happens for p -adic fractal strings, the volume V L p ( ε ) will be acontinuous function of ε in this context. (Compare with Remark 4.8.) We expect that thehigher-dimensional tube formulas obtained by Lapidus and Pearse in [ , ] (aswell as, more generally, by those same authors and Winter in [ ]) for archime-dean self-similar sprays and the associated tilings [ ] in R d have a natural nonar-chimedean counterpart in the d -dimensional p -adic space Q dp , for any integer d ≥ p -adic case, the corresponding ‘tubular zeta function’ ζ T p ( ε ; s ) (when d = 1, see Remark 6.2) should have a more complicated expression than in the one-dimensional situation, and should involve both the inner radii and the ‘curvature’ ofthe generators (see [ , ], as described in [ , § p -adic fractal spray) T p . Moreover, by analogy with what is expectedto happen in the Euclidean case [ , ], the coefficients of the resulting higher-dimensional tube formula should have an appropriate interpretation in terms ofyet to be suitably defined ‘nonarchimedean fractal curvatures’ associated with eachcomplex and integral dimension of T p . Finally, by analogy with the archimedeancase (for d ≥
1, see [ ] and [ ]), the p -adic higher-dimensional fractal tube for-mula should take the same form as in Equation (6.6), except with ζ L p ( ε ; s ) given bya different expression from the one in (6.7) where d = 1, and with D L p ( W ) replacedby D L p ( W ) ∪ { , , . . . , d } , as well as (for nonarchimedean self-similar tilings) with W = C and R p ( ε ) ≡ § R N of any dimension and for arbitrarybounded subsets of R N (see, e.g., the book [ ]). In the process, they have verysignificantly extended the theory of fractal tube formulas obtained originally forfractal strings in [ , ] and then for higher-dimensional fractal sprays (especially,self-similar sprays) in [ – ]; see, especially, [ , ] and [ , Ch. 5]. Accordingly,it is natural to wonder whether the general theory of fractal zeta functions andfractal tube formulas developed in [ ] and [ – ] can be applied and suitablyadapted in order to obtain concrete nonarchimedean tube formulas valid (underappropriate hypotheses) for arbitrary compact subsets of p -adic space ( Q p ) N (ormore general ultrametric spaces), and, in particular, for arbitrary p -adic self-similarsets in ( Q p ) N .
8. Epilogue
In looking for a simple geometric way to create an ad`elic fractal string and aglobal theory of complex fractal dimensions, we found a very natural constructionof p -adic fractal strings of any rational dimension between 0 and 1. The simplest INKOWSKI DIMENSION AND TUBE FORMULAS FOR p -ADIC FRACTAL STRINGS 29 example is of dimension D = , which is particularly interesting since it involvesthe diagonal of the digits. This reminds one of the intersections of the graph of theFrobenius with the diagonal in Enrico Bombieri’s proof of the Riemann hypothesisfor curves over finite fields; see [ , ]. It may give rise to a fractal approachto translating his proof for curves over finite fields to the curve spec Z over therationals, which is the case of the famous Riemann hypothesis for the Riemannzeta function. However, we caution the reader that this possibility is far from beingrealized for now.We found another natural way to create an infinite family of p -adic Cantorstrings CS p in the nonarchimedean ring of p -adic integers Z p and simultaneouslytheir exact counterparts in the archimedean unit interval [0 , p -inary Cantorstrings CS ∗ p . The Minkowski dimensions of the nonarchimedean and archimedeanCantor strings vary from 0 to 1 as p varies from 2 to ∞ . Directly above and belowthe Minkowski dimension lie infinitely many complex fractal dimensions, period-ically distributed along a discrete vertical line. The periodic distribution of thecomplex fractal dimensions, being discrete near dimension 0, become denser as theMinkowski dimension tends to 1.The simplest way to unify all infinitely many p -adic Cantor strings CS p togetherwith the ordinary real Cantor string CS is to form an infinite product CS × Y p< ∞ CS p , which is a self-similar string in the set of integral ad`eles A Z .An even more harmonious and symmetric way to unify all the nonarchimedeanCantor strings together with their corresponding archimedean counterpart is firstto pair each p -adic Cantor string CS p together with the p -inary Cantor string CS ∗ p by taking the Cartesian product CS p × CS ∗ p ⊂ Q p × R . Then we can imagine theinfinite direct product Y p< ∞ ( CS p × CS ∗ p )as being an ‘ad`elic’ Cantor string in a new ‘ad`elic’ space Y p< ∞ ( Q p × R )with infinitely many archimedean components. We note, however, that our constructions of ad`elic fractal strings do not givethe Riemann zeta function as the geometric zeta function. It would be interestingto have a natural construction of an ad`elic fractal string with the Riemann zetafunction as its geometric zeta function.We conclude these comments with a construction that gives the square of theRiemann zeta function. Let E p be the p -adic Euler string and h be the real harmonicstring, then the infinite direct product h × Y p< ∞ E p We put ‘ad`elic’ in quotes because this space has infinitely many real components, one forevery prime number, whereas the ring of ad`eles has only one real component, corresponding tothe archimedean valuation of Q . can be considered as an ad`elic fractal string in the set of integral ad`eles A Z . Let ζ E p be the geometric zeta function of E p and ζ h be the geometric zeta function ofthe harmonic string; then, the infinite product of complex meromorphic functions ζ h × Y p< ∞ ζ E p is equal to the square of the Riemann zeta function. Acknowledgments
We wish to thank Springer, the publisher of [ ], for having granted us (morespecifically, the two authors of the book [ ]) the copyright for [ , Section 13.2]within which part of the results obtained in this paper were discussed. That portionof [ , Section 13.2] was referring, in particular, to an earlier preprint of this articleto which we have since then made a number of changes and additions, includingthe new result providing (in § p -adic (or nonarchimedean) case. Another significantaddition to our earlier version of this paper is the discussion (in §
8) of several resultsconcerning the new topic of ad`elic fractal strings, for which complete proofs will beprovided in a later work towards a global theory of complex fractal dimensions.
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