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

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## Abstract

**:**

Nature is an infinite sphere of which the center is everywhere and the circumference nowhere.Blaise Pascal (1623–1662)

## 1. Introduction

The inverse spectral problem for a fractal string can be solved if and only if its dimension is not $1/2$.

## 2. Nonarchimedean Fractal Strings

#### 2.1. p-Adic Numbers

**Remark**

**1.**

- (a)
- The distance ${d}_{p}$ defined on ${\mathbb{Q}}_{p}$ by ${d}_{p}(x,y)={|x-y|}_{p}$ is called an ultrametric, since it satisfies the counterpart of the above strong triangle inequality:$${d}_{p}(x,z)\le max\{{d}_{p}(x,y),{d}_{p}(y,z)\}$$$$\begin{array}{c}If\phantom{\rule{4.pt}{0ex}}{d}_{p}(x,y)>{d}_{p}(y,z)\phantom{\rule{4.pt}{0ex}}\mathit{then}\phantom{\rule{4.pt}{0ex}}{d}_{p}(x,z)={d}_{p}(x,y).\end{array}$$It follows that the center can be chosen anywhere within the p-adic ball B. Moreover, given any two balls ${B}_{1}$ and ${B}_{2}$, either they are disjoint or one is entirely contained in the other (i.e., ${B}_{1}\subseteq {B}_{2}$ or ${B}_{2}\subseteq {B}_{1}$). These special properties are common to all ultrametric spaces (i.e., all metric spaces for which the ultrametric triangle inequality (2) holds).
- (b)
- By definition, ${\mathbb{Z}}_{p}$ is the (closed) unit ball of (${\mathbb{Q}}_{p},{d}_{p}$). Moreover, ${\mathbb{Z}}_{p}$ has the remarkable property of being a ring (since for all $x,y$ in ${\mathbb{Z}}_{p}$, by (2) again, ${|x+y|}_{p}\le {max\left(\right|x|}_{p}{,\left|y\right|}_{p})\le 1$, and ${\left|xy\right|}_{p}={\left|x\right|}_{p}{\left|y\right|}_{p}\le 1$). This is to be contrasted with the fact that $[-1,1]$, the unit ball of $\mathbb{R}$, is not stable under addition (although it is obviously stable under multiplication); see [61]. Finally, since translations are homeomorphisms, every closed ball $B=B(a,r)$ in ${\mathbb{Q}}_{p}$ with center a has a radius r of the form $r={p}^{n}$,$$B(a,r)=a+{p}^{-n}{\mathbb{Z}}_{p}=\{x\in {\mathbb{Q}}_{p}{:|x-a|}_{p}\le r\}$$
- (c)
- (p-adic intervals). In the sequel (as well as in part of the literature on p-adic analysis, see, e.g., [55]), the metric balls $B=a+r{\mathbb{Z}}_{p}$ (with $a\in {\mathbb{Q}}_{p}$ and $r\in {p}^{\mathbb{Z}}$, as in (4) just above), are sometimes called the ‘intervals’ of ${\mathbb{Q}}_{p}$. Note that they are not connected, in the usual topological sense, but that they are ‘convex’, in the following sense: for each $x,y\in B$ and $\alpha \in {\mathbb{Z}}_{p}$, we have that $\alpha x+(1-\alpha )y\in B$. (Here and henceforth, it is useful to think of ${\mathbb{Z}}_{p}\subset {\mathbb{Q}}_{p}$ as being the analogue of the unit interval $[0,1]\subset \mathbb{R}$, rather than of $[-1,1]$.)
- (d)
- (The archimedean/nonarchimedean dichotomy). A beautiful and classical theorem of Alexander Ostrowski states that each nontrivial absolute value on the field of rational numbers $\mathbb{Q}$, is either equivalent to the standard archimedean absolute value on $\mathbb{Q}$ or to the nonarchimedean p-adic absolute value ${|\xb7|}_{p}$ for some prime p. (Recall that two absolute values are said to be equivalent if they induce the same topology on $\mathbb{Q}$; this is the case if and only if one is a power of the other.) Therefore, infinitely many completions of $\mathbb{Q}$ (one for each prime p) are nonarchimedean and $\mathbb{R}$ is the only completion of $\mathbb{Q}$ that is archimedean. For this reason, one sometimes writes $\mathbb{R}={\mathbb{Q}}_{\infty}$ and refers to (the equivalence class of) the absolute value $|\xb7|$ as the ‘place at infinity’, associated with the ‘prime at infinity’ or the ‘real prime’; see [61]. (We note that Ostrowski’s Theorem is usually expressed in terms of valuations rather than of absolute values. Accordingly, a place of $\mathbb{Q}$ is generally defined as an equivalence class of valuations on $\mathbb{Q}$.) With this notation in mind, we see that the field ${\mathbb{Q}}_{\infty}$ is archimedean, whereas for any (finite) prime p, ${\mathbb{Q}}_{p}$ is a nonarchimedean field. The theory of p-adic fractal strings developed in [30,31,32,33] is aimed, initially, at finding suitable definitions and obtaining results that parallel those corresponding to the theory of real (or archimedean) fractal strings developed in [7], for example. As we will see, however, although there are many analogies between the archimedean and nonarchimedean theories of fractal strings, there are also some notable differences between them; see, especially, [32], along with [30,33].

#### 2.2. p-Adic Fractal Strings

**Definition**

**1.**

**Remark 2**(Convex components)

**.**

**Definition**

**2.**

**Definition**

**3.**

**Remark**

**3.**

#### 2.3. Example: p-Adic Euler String

**Remark 4**(The punctured unit ball)

**.**

**Remark 5**(Adèlic Euler string)

**.**

**Remark 6**(Comparison with the archimedean theory)

## 3. The Geometric Zeta Function

**Definition**

**4.**

**Remark**

**7.**

**Remark 8**(Archimedean fractal strings)

**.**

#### Languid and Strongly Languid p-Adic Fractal Strings

**Definition**

**5.**

- $\mathbf{L}\mathbf{1}$ For all $n\in \mathbb{Z}$ and all $u\ge S\left({T}_{n}\right),$$$|{\zeta}_{{\mathcal{L}}_{p}}(u+i{T}_{n})|\le C(|{T}_{n}{|+1)}^{\kappa},$$
- $\mathbf{L}\mathbf{2}$ For all $t\in \mathbb{R},\left|t\right|\ge 1,$$$|{\zeta}_{{\mathcal{L}}_{p}}{(S\left(t\right)+it)|\le C|t|}^{\kappa}.$$

- $\mathbf{L}{\mathbf{2}}^{\prime}$ There exist constants $A,C>0$ such that for all $t\in \mathbb{R}$ and $m\ge 1$,$$|{\zeta}_{{\mathcal{L}}_{p}}({S}_{m}\left(t\right)+it)|\le C{A}^{|{S}_{m}\left(t\right)|}{\left(\right|t|+1)}^{\kappa}.$$

**Remark**

**9.**

- (a)
- Intuitively, hypothesis $\mathbf{L}\mathbf{1}$ is a polynomial growth condition along horizontal lines (necessarily avoiding the poles of ${\zeta}_{{\mathcal{L}}_{p}}$), while hypothesis $\mathbf{L}\mathbf{2}$ is a polynomial growth condition along the vertical direction of the screen.
- (b)
- Clearly, condition $\mathbf{L}{\mathbf{2}}^{\prime}$ is stronger than $\mathbf{L}\mathbf{2}$. Indeed, if ${\mathcal{L}}_{p}$ is strongly languid, then it is also languid (for each screen ${S}_{m}$ separately).
- (c)
- Moreover, if ${\mathcal{L}}_{p}$ is languid for some κ, then it is also languid for every larger value of κ. The same is also true for strongly languid strings.
- (d)
- Finally, hypotheses $\mathbf{L}\mathbf{1}$ and $\mathbf{L}\mathbf{2}$ require that ${\zeta}_{{\mathcal{L}}_{p}}$ has an analytic (i.e., meromorphic) continuation to an open, connected neighborhood of $\Re \left(s\right)\ge {\sigma}_{{\mathcal{L}}_{p}}$, while $\mathbf{L}{\mathbf{2}}^{\prime}$ requires that ${\zeta}_{{\mathcal{L}}_{p}}$ has a meromorphic continuation to all of $\mathbb{C}$.

## 4. Volume of Thin Inner Tubes

**Definition**

**6.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**1**(Volume of thick inner tubes)

**.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Definition**

**7.**

**Theorem 2**(Volume of thin inner tubes)

**.**

**Proof.**

**Remark**

**10.**

**Remark 11**(Comparison between the archimedean and the nonarchimedean cases).

#### 4.1. Example: The Euler String

## 5. Minkowski Dimension

**Remark**

**12.**

**Theorem**

**3.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**2.**

#### 5.1. The Real Case

**Remark**

**13.**

## 6. Explicit Tube Formulas for $\mathbf{p}$-adic Fractal Strings

**Theorem 5**(p-adic explicit tube formula)

**.**

- (i)
- Let ${\mathcal{L}}_{p}$ be a languid p-adic fractal string (as in the first part of Definition 5 of Section 3), for some real exponent κ and a screen S that lies strictly to the left of the vertical line $\Re \left(s\right)=1$. Further assume that ${\sigma}_{{\mathcal{L}}_{p}}<1.$(Recall from Corollary 2 that we always have ${\sigma}_{{\mathcal{L}}_{p}}\le 1$. Moreover, if ${\mathcal{L}}_{p}$ is self-similar, then ${\sigma}_{{\mathcal{L}}_{p}}<1.)$ Then the volume of the thin inner ε-neighborhood of ${\mathcal{L}}_{p}$ is given by the following distributional explicit formula, on test functions in $\mathbf{D}(0,\infty )$, the space of ${C}^{\infty}$ functions with compact support in $(0,\infty ):$$${V}_{{\mathcal{L}}_{p}}\left(\epsilon \right)=\sum _{\omega \in {\mathcal{D}}_{{\mathcal{L}}_{p}}\left(W\right)}\mathrm{res}\left(\frac{{p}^{-1}{\zeta}_{{\mathcal{L}}_{p}}\left(s\right){\epsilon}^{1-s}}{1-s};\omega \right)+{\mathcal{R}}_{p}\left(\epsilon \right),$$$${\mathcal{R}}_{p}\left(\epsilon \right)=\frac{1}{2\pi i}{\int}_{S}\frac{{p}^{-1}{\zeta}_{{\mathcal{L}}_{p}}\left(s\right){\epsilon}^{1-s}}{1-s}ds$$$${\mathcal{R}}_{p}\left(\epsilon \right)=O\left({\epsilon}^{1-supS}\right),\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{3.33333pt}{0ex}}\epsilon \to {0}^{+}.$$
- (ii)
- Moreover, if ${\mathcal{L}}_{p}$ is strongly languid (as in the second part of Definition 5), then we can take $W=\mathbb{C}$ and ${\mathcal{R}}_{p}\left(\epsilon \right)\equiv 0$, provided we apply this formula to test functions supported on compact subsets of $[0,A)$. The resulting explicit formula without error term is often called an exact tube formula in this case.

**Proof.**

**Remark**

**14.**

**Corollary 3**(p-adic fractal tube formula).

**Remark**

**15.**

**Example 1**(Fractal tube formula for the p-adic Euler string)

**.**

**Example 2**(The tube formula for the nonarchimedean Cantor string)

**.**

**Remark**

**16.**

- (i)
- Because on each relevant vertical line, the complex dimensions form an arithmetic progression (with a progression or period independent of the line) and have the same multiplicities, the corresponding term in the associated fractal tube formula can be written as a suitable power function times a periodic function (of $x:=\mathrm{log}\left({\epsilon}^{-1}\right)$). (This is so assuming that the complex dimensions on that line are simple, which is always the case, for instance, of the right most vertical line $\{\mathfrak{R}s={D}_{M}\}$).
- (ii)
- In all of the concrete examples of p-adic self-similar strings studied in [32,33], 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 5). We conjecture that at least in the case of simple complex dimensions, the exact fractal tube formula of a p-adic self-similar string always converges pointwise (and not just distributionally, as in Theorem 5). (Such a result is established in [7], Section 8.4 for general real or archimedean self-similar strings, whether or not all of the complex dimensions are simple.) Accordingly, it would be very interesting to establish that conjecture as well as to obtain a pointwise counterpart of Theorem 5; that is, a fractal tube formula for p-adic (not necessarily self-similar) fractal strings, with or without an error term, which (under suitable hypotheses) would be valid pointwise. We note that in the archimedean case (i.e., for real fractal strings) such a pointwise fractal tube formula is available under rather general conditions; see [7], Section 8.1.1, esp., Theorem 8.7 and Corollary 8.10. We leave the investigation of these issues to some future work or to the interested reader.

## 7. Possible Extensions

#### 7.1. Adèlic Fractal Strings and Their Spectra

#### 7.2. Nonarchimedean Fractal Strings in Berkovich Space

#### 7.3. Higher-Dimensional Fractal Tube Formulas

## 8. Epilogue

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Lapidus, M.L.; Lũ’, H.; Van Frankenhuijsen, M.
Minkowski Dimension and Explicit Tube Formulas for *p*-Adic Fractal Strings. *Fractal Fract.* **2018**, *2*, 26.
https://doi.org/10.3390/fractalfract2040026

**AMA Style**

Lapidus ML, Lũ’ H, Van Frankenhuijsen M.
Minkowski Dimension and Explicit Tube Formulas for *p*-Adic Fractal Strings. *Fractal and Fractional*. 2018; 2(4):26.
https://doi.org/10.3390/fractalfract2040026

**Chicago/Turabian Style**

Lapidus, Michel L., Hùng Lũ’, and Machiel Van Frankenhuijsen.
2018. "Minkowski Dimension and Explicit Tube Formulas for *p*-Adic Fractal Strings" *Fractal and Fractional* 2, no. 4: 26.
https://doi.org/10.3390/fractalfract2040026