# New 5-adic Cantor sets and fractal string

- Ashish Kumar
^{1}Email author, - Mamta Rani
^{2}and - Renu Chugh
^{1}

**Received: **1 October 2013

**Accepted: **22 November 2013

**Published: **5 December 2013

## Abstract

In the year (1879–1884), George Cantor coined few problems and consequences in the field of set theory. One of them was the Cantor ternary set as a classical example of fractals. In this paper, 5-adic Cantor one-fifth set as an example of fractal string have been introduced. Moreover, the applications of 5-adic Cantor one-fifth set in string theory have also been studied.

## Introduction

During the late eighteenth century, mathematicians delighted in producing sets with ever more weird properties, many of them now recognized to be fractal in nature (Crilly et al.). George Cantor (1879–1884) wrote a series of papers entitled “Uber unendliche lineare punktmannichfaltigkeiten” (Cantor 1879; 1880; 1882; 1883a; 1883b; 1884) that contained the first systematic treatment of the point set topology of real line, in which he triggered some problems and consequences in the field of set theory. One of these is the classical Cantor set problem devised by Cantor in the footnote to a statement saying that perfect sets do not need to be everywhere dense (Fleron 1994). In last two decades, Devil’s and other researchers established the graphical representation of Cantor sets in the form of staircases (Horiguchi and Morita 1984a; 1984b; Rani and Prasad 2010).

Middle one-third, a classical Cantor set found a celebrated place in the mathematical analysis and in its applications (Hutchinson 1981; Mendes 1999; Shaver 2010). For a fundamental work on Cantor set and its applications, one may refer to (Peitgen et al. 2004), (Devaney 1992), (Beardon 1965), (Falconer 1985), (Lapidus and van Frankenhuijsen 2006), (Gutfraind et al. 1990) and (Lee 1998). In recent years, p-adic analysis has been used in various areas of mathematics as well as in aspects of quantum physics and string theory (Lapidus and van Frankenhuijsen 2006). For a detailed analysis of fractal string and p-adic integers, one may refer to (Chistyakov 1996; Hung 2007; Koblitz 1984; Robert 2000; Schikhof 1984; Vladimirov et al. 1994).

Lapidus and van Frankenhuijsen (2000; 2006) introduced the concept of fractal string and established the geometric zeta function, zeros of zeta function, spectra of fractal string and the complex dimension of the fractal string. In 2008, (Lapidus 2008) suggested that fractal string and their quantization may be related to aspects of string theory. In last few decades, M. L. Lapidus, jointly with other researchers generalized and introduced the various properties of fractal string (see (Edgar 2008; Lapidus 1992; Lapidus and Maier 1995; Lapidus and Pearse 2006; 2008; Lapidus and Pomerance 1993)).

In 2008, (Lapidus and Hung 2008; 2009) provided a framework for unifying the archimedean and p-adic (nonarchimedean) fractal string with their geometric zeta functions and complex dimensions for 3-adic Cantor sets and also the general case for p-adic Cantor sets respectively. Recently, (Ashish, Mamta Rani and Renu Chugh, Variants of Cantor Sets Using IFS, submitted and Ashish, Mamta Rani and Renu Chugh, Study of Variants of Cantor sets., submitted) studied the variants of Cantor sets and established their mathematical analysis using mathematical feedback system and iterated function system respectively.

Our goal in this paper is to study the Cantor one-fifth set as a new classical example of fractal string. Moreover, the non-archimedean (5-adic) Cantor one-fifth set with their applications in string theory has also been established. In the third section, the main results of our study have been presented, followed by the “Concluding remarks” section.

## Preliminaries

In this section, we recall some basic definitions pertaining to the notion of (ordinary) fractal string and introduce several new ones such as the most important of which are quinary expansion and Cantor one-fifth set:

### Definition 2.1. Cantor one-fifth set

The Cantor one-fifth set for unequal intervals is defined as the *F* = ∩ *F*_{n+1}, where *F*_{n+1} is constructed by dividing *F*_{
n
} in five unequal line segments and removing second and fourth one-fifth line segment, *F*_{0} being the closed interval 0 ≤ *x* ≤ 1 (Ashish, Mamta Rani and Renu Chugh, Variants of Cantor Sets Using IFS, submitted).

### Definition 2.2. Quinary expansion

The sequence 0.*x*_{1}*x*_{2}*x*_{3}*x*_{4}*x*_{5}…, where each *x*_{
i
} is either 0, 1, 2, 3, or 4 is called quinary expansion of *x* if *x* = *x*_{1}/5 + *x*_{2}/5^{2} + *x*_{3}/5^{3} + ....

Lapidus and van Frankenhuijsen (2000) and (2006), introduced the concept of fractal strings as follows:

### Definition 2.3. Fractal string

A fractal string Ω is a bounded open subset of the real line *R*. The collection of lengths ℓ_{
j
} of the disjoint intervals is denoted by *L*.

For example, the complement of the Cantor set in the closed unit interval [0, 1] is a Cantor string. Moreover, the topological boundary of Cantor string is the Cantor set *C* itself.

### Definition 2.4. Geometric zeta function

*L*is

where ℓ_{1}, ℓ_{2}, …, ℓ_{
k
} are the lengths of open intervals and *m*_{
k
} be the corresponding multiplicity of open intervals (Lapidus and van Frankenhuijsen 2000).

_{1}= (

*l*

_{1}= 1/3), ℓ

_{2}= (

*l*

_{2}=

*l*

_{3}= 1/9), ℓ

_{3}= (

*l*

_{4}=

*l*

_{5}=

*l*

_{6}=

*l*

_{7}= 1/27), and so on, that is, the lengths are the numbers 3

^{-k-1}with multiplicity ${m}_{{3}^{-k-1}}={2}^{k}$ for

*k*= 0, 1, 2, 3, …. . So, the geometric zeta function is:

where D = log2/log3 is the dimension of usual Cantor set.

Recently, (Ashish, Mamta Rani and Renu Chugh, Variants of Cantor Sets Using IFS, submitted), established the self-similarity of the Cantor one-fifth set using the iteration function system as follows:

### Theorem 2.1

*f*

_{1},

*f*

_{2}and

*f*

_{3}be the similarity contraction mappings on

*ℝ*defined by

*F*satisfies the self-referential equation

for the iterated function system (*f*_{1}, *f*_{2}, *f*_{3}).

## Main results

### 5-adic (nonarchimedean) Cantor one-fifth set

*s*

_{ i })

_{i ∈ ℕ}of natural numbers between 0 and

*p*-1 (inclusive) is a

*p*-adic integer. We write this conventionally as .....

*s*

_{ i }.....

*s*

_{2}

*s*

_{1}

*s*

_{0}. If ‘

*n*’ is any natural number, and

*p*-adic representation (in other words, $n={\displaystyle {\sum}_{i=0}^{k-1}{s}_{i}}{p}^{i}$ with each

*s*

_{ i }is a

*p*-adic digit), then we identify ‘

*n*’ with the

*p*-adic integer (

*s*

_{ i }) with

*s*

_{ i }= 0 if

*i*≥

*k*(Madore 2000). Further, the set of

*p*-adic integers, which we call

*ℤ*

_{ p }with two binary operations on it (addition and multiplication) is a ring. The relation between the set (ring)

*ℤ*

_{ p }of

*p*-adic integers and the set (field)

*ℚ*

_{ p }of

*p*-adic numbers is the same as between the set (ring)

*ℤ*of integers and the set (field)

*ℚ*of rationals (Madore 2000). Since,

*ℤ*

_{ p }is an important subspace of

*ℚ*

_{ p }, it can be represented as follows:

*p*-adic expansion, we can also write

where *c* + *pℤ*_{
p
} = {*y* ∈ *ℚ*_{
p
} : |*y* - *c*|_{
p
} ≤ 1/*p*} (Lapidus and van Frankenhuijsen 2006) It is also known that there are topological models of *ℤ*_{
p
} in the Euclidean space *ℝ*^{
d
} as fractal spaces such as the Cantor set and the Sierpinsky gasket (Robert 2000), where *ℤ*_{
p
} is homeomorphic to the ternary Cantor set. Now, we consider the ring of 5-adic integers *ℤ*_{5}, that is, homeomorphic to Cantor one-fifth set.

*N*’. To start the construction, initiator

*N*

_{0}=

*ℤ*

_{5}is subdivided into five equal subintervals 0 + 5

*ℤ*

_{5}, 1 + 5

*ℤ*

_{5}, 2 + 5

*ℤ*

_{5}, 3 + 5

*ℤ*

_{5}and 4 + 5

*ℤ*

_{5}. Drop the subintervals 1 + 5

*ℤ*

_{5}and 3 + 5

*ℤ*

_{5}and repeat the same process for the remaining subintervals. Further, repeating the same process over and over again, by removing the open subintervals of second and fourth position at each step from each closed interval, we obtain a sequence

*N*

_{ k }for

*k*= 1, 2, . . . The 5-adic Cantor one-fifth set (see Figure 1)

*N*

_{ k }consists of 3

^{ k }disjoint closed intervals. Thus, the 5-adic Cantor one-fifth set would be the limit ‘

*N*’ of the sequence

*N*

_{ k }of sets. So, we define limit ‘

*N*’ as the intersection of the sets

*N*

_{ k }i.e.

#### Theorem 3.1

*f*

_{1},

*f*

_{2}and

*f*

_{3}be the similarity contraction mappings on 5-adic integer

*ℤ*

_{5}defined by

*N*satisfies the self-referential equation

for all *k* ≥ 1. Since, the mapping *f*_{
j
} for *j* = 1, 2, 3 is one-to-one and *N* = ∩ *N*_{
k
}, then it implies that

*f*_{
j
}[*N*] = *f*_{
j
}[ ∩ *N*_{
k
}] = ∩ *f*_{
j
}[*N*_{
k
}], for *k* = 1, 2, ….

so that, we can write *f*_{1}[*N*] = ∩ *f*_{1}[*N*_{
k
}], *f*_{2}[*N*] = ∩ *f*_{2}[*N*_{
k
}]and *f*_{3}[*N*] = ∩ *f*_{3}[*N*_{
k
}],

*f*

_{1}[

*N*] ∪

*f*

_{2}[

*N*] ∪

*f*

_{3}[

*N*] = ( ∩

*f*

_{1}[

*N*

_{ k }]) ∪ ( ∩

*f*

_{2}[

*N*

_{ k }]) ∪ ( ∩

*f*

_{3}[

*N*

_{ k }])

which gives the proof of the theorem.

*f*

_{1},

*f*

_{2},

*f*

_{3}).

### Quinary expansion of 5-adic Cantor one-fifth set

#### Theorem 3.2

for all *j* = 0, 1, 2, .....

*f*

_{1},

*f*

_{2}and

*f*

_{3}, on

*ℤ*

_{5}as follows:

*x*

_{ j }∊ {0, 1, 2, 3, 4}, for all

*j*≥ 0, either

*x*

_{0}= 1 or

*x*

_{0}= 3, respectively. Let

*η*,

*μ*∈

*ℕ*be the fixed subscript numbers such that

*x*

_{ η }= 1and

*x*

_{ μ }= 3. Thus,

*x*

_{ j }= 0, 2 or 4, for all

*j*>

*η*and all

*j*>

*μ*. Since, we have divided the real line into five equal line segments denoted by 0, 1, 2, 3, and 4 respectively. Thus, if

*x*

_{0}= 0, then we use the function

*f*

_{1}

^{-1}for all

*x*∊

*N*, if

*x*

_{0}= 2, then use the function

*f*

_{2}

^{-1}for all

*x*∊

*N*and if

*x*

_{0}= 4, then use the function

*f*

_{3}

^{-1}for all

*x*∊

*N*. Thus, from these three cases, we obtain

*ℤ*

_{5}and 3 + 5

*ℤ*

_{5}respectively. Thus, we found that

Hence either *x* ∈ 1 + 5*ℤ*_{5} or *x* ∈ 3 + 5*ℤ*_{5} which deduce that *x* ∉ *N*. Hence we proved that for *x*_{
j
} ∊ {0, 2, 4}, *x* ∊ *N*.

*x*=

*x*

_{0}+ 5

^{1}

*x*

_{1}+ 5

^{2}

*x*

_{2}+ …, belong to

*ℤ*

_{5}for all

*x*

_{ j }∊ {0, 2, 4}, and

*j*= 0, 1, 2, …. Then, from Eq. (3) and (5), we can say that neither

*x*∈ 1 + 5

*ℤ*

_{5}nor

*x*∈ 3 + 5

*ℤ*

_{5}which implies that

*x*∉

*f*

_{ j }(1 + 5

*ℤ*

_{5}) and also

*x*∉

*f*

_{ j }(3 + 5

*ℤ*

_{5}), for

*j*∊

*W*

_{ l }= {1, 2, 3}

^{ l },

*l*= 0, 1, 2, ..... Thus,

Thus, *N* ∪ *Y* = *ℤ*_{5} and hence *x* ∊ *N*, which completes the proof of the theorem.

### Cantor one-fifth set as fractal string

*L*= ℓ

_{1}, ℓ

_{2}, ℓ

_{3}, …, called the lengths of the string. We can assume without loss of generality that

where each length is counted according to its multiplicity. An ordinary fractal string can be thought of as a one-dimensional drum with fractal boundary. In the literature of fractal geometry, we found a classical example of the fractal string as Cantor string. It is the set, complement of the interval [0, 1] of the usual ternary Cantor set. It is one of the simplest and most important example in the research of fractal string by (Lapidus and van Frankenhuijsen 2006). Information about the geometry of Cantor string like Minkowski dimension and the Minkowski measurability is obtained from its geometric zeta function. Motivated by the research of Lapidus with other researcher’s (Lapidus and Hung 2008) on the Cantor string, we introduce a new Cantor one-fifth set as an example of fractal string.

*F*. The Figure 3 shows the geometrical representation of Cantor one-fifth string.

where, ℓ_{1} = (*l*_{1} = *l*_{2} = 1/5), ℓ_{2} = (*l*_{3} = *l*_{4} = *l*_{5} = *l*_{6} = *l*_{7} = *l*_{8} = 1/25) and so on. Continuing in this way, we find that the lengths of open intervals is consist of ℓ_{
k
} = 5^{-k-1} with multiplicity ${m}_{{5}^{-k-1}}={2.3}^{k}$ for *k* = 0, 1, 2, ....

where *D* = log 3/log 5 = 0.6826 is the dimension of Cantor one-fifth set and *p* = 2*π*/log 5 oscillatory period of Cantor one-fifth string ℵ, is called *complex dimension* of Cantor one-fifth string.

### 5-adic Cantor one-fifth set as fractal string

*ξ*) is analogue to the usual Cantor one-fifth set. We start, by subdividing the interval

*ℤ*

_{5}into closed subintervals

*G*

_{1}∪

*G*

_{2}is the first sub-ring of self similar 5-adic Cantor one-fifth string. The lengths of

*G*

_{1}and

*G*

_{2}are given by using the Haar measure (Gupta and Jain 1986) as follows:

*G*

_{3}∪

*G*

_{4}∪

*G*

_{5}∪

*G*

_{6}∪

*G*

_{7}∪

*G*

_{8}is the second set of self-similar 5-adic Cantor one-fifth string. Thus, the length is given by

_{1}= ℓ

_{2}= ℓ

_{3}= ℓ

_{4}= ℓ

_{5}= ..... which consists of lengths 5

^{-k-1}with multiplicity 2.3

^{ k }. Using Figure 5 the 5-adic Cantor one-fifth string can also be written as follows:

*ξ*is given by

where *D* = log 3/log 5 = 0.6826 is the dimension of 5-adic Cantor one-fifth string and *p* = 2*π*/log 5 oscillatory period is the volume of the inner tubular neighborhood of *ξ*.

## Concluding remarks

- 1.
In Subsection “5-adic (nonarchimedean) Cantor one-fifth set”, using 5-adic integer it has been concluded that Cantor one-fifth set satisfies the nonarchimedean properties of a set and also studied that nonarchimedean Cantor one-fifth set satisfies self-similarity property using self-referential equation.

- 2.
Further, it has been concluded that quinary Cantor one-fifth set is homeomorphic to 5-adic Cantor one-fifth set

*N*in subsection “Quinary expansion of 5-adic Cantor one-fifth set”. - 3.
In Subsection “Cantor one-fifth set as fractal string” and “5-adic Cantor one-fifth set as fractal string”, it has been analyzed that Cantor one-fifth set and 5-adic Cantor one-fifth set both satisfy the properties of fractal string. Moreover, we found that the geometric zeta function and the complex dimension of both the sets are perfectly same.

## Declarations

### Acknowledgments

This Research is supported by the University Grant Commission of India (Grant No. 39-29/2010(SR)).

## Authors’ Affiliations

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