- Research
- Open Access

# 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.

- Cantor one-fifth set
- p-adic integers
- 5-adic Cantor one-fifth set
- Iterated function system (IFS)
- Fractal string

- 26A30
- 28A80
- 11E20
- 26E30
- 28E30
- 26A80
- 28A12

## 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

## References

- Beardon AF: On the Hausdorff dimension of general Cantor sets.
*Proc Camb Phil Soc*1965, 61: 679-694. 10.1017/S0305004100039049View ArticleGoogle Scholar - Cantor G: Uber unendliche lineare Punktmannichfaltigkeiten, Part 1.
*Math Ann*1879, 15: 1-7. 10.1007/BF01444101View ArticleGoogle Scholar - Cantor G: Uber unendliche lineare Punktmannichfaltigkeiten, Part 2.
*Math Ann*1880, 17: 355-358. 10.1007/BF01446232View ArticleGoogle Scholar - Cantor G: Uber unendliche lineare Punktmannichfaltigkeiten, Part 3.
*Math Ann*1882, 20: 113-121. 10.1007/BF01443330View ArticleGoogle Scholar - Cantor G: Uber unendliche lineare Punktmannichfaltigkeiten, Part 4.
*Math Ann*1883, 21: 51-58. 10.1007/BF01442612View ArticleGoogle Scholar - Cantor G: Uber unendliche lineare Punktmannichfaltigkeiten, Part 5.
*Math Ann*1883, 21: 545-591. 10.1007/BF01446819View ArticleGoogle Scholar - Cantor G: Uber unendliche lineare Punktmannichfaltigkeiten, Part 6.
*Math Ann*1884, 23: 453-488. 10.1007/BF01446598View ArticleGoogle Scholar - Chistyakov DV: Fractal geometry of continuous embeddings of p-adic numbers into Euclidean spaces.
*Theor Math Phys*1996, 109: 1495-1507. 10.1007/BF02073866View ArticleGoogle Scholar - Crilly AJ, Earnshaw RA, Jones H:
*Fractal and Chaos*. New York: Springer-Verlag; 1991.View ArticleGoogle Scholar - Devaney RL:
*A First Course in Chaotic Dynamical Systems*. Holland: Addison Wesley Pub. Company, Inc; 1992:75-79.Google Scholar - Edgar G:
*Measure, Topology, and Fractal Geometry*. New York: Springer Verlag; 2008.View ArticleGoogle Scholar - Falconer K:
*The Geometry of Fractal Sets*. Cambridge: Cambridge University Press; 1985.View ArticleGoogle Scholar - Fleron JF: A note on the history of the Cantor set and Cantor functions.
*Math Mag*1994, 67: 136-140. 10.2307/2690689View ArticleGoogle Scholar - Gupta VP, Jain PK:
*Lebesgue Measure and Integration*. New Delhi: John Wiley & Sons; 1986.Google Scholar - Gutfraind R, Sheintuch M, Avnir D: Multifractal scaling analysis of diffusion-limited reactions with Devil’s staircase and Cantor set catalytic structures.
*Chem Phys Lett*1990, 174(1):8-12. 10.1016/0009-2614(90)85318-7View ArticleGoogle Scholar - Horiguchi T, Morita T: Devil’s staircase in one dimensional mapping.
*Physica A*1984, 126(3):328-348. 10.1016/0378-4371(84)90205-XView ArticleGoogle Scholar - Horiguchi T, Morita T: Fractal dimension related to Devil’s staircase for a family of piecewise linear mappings.
*Physica A*1984, 128(1–2):289-295.View ArticleGoogle Scholar - Hutchinson JE: Fractals and self-similarity.
*Indiana Univ Math J*1981, 30: 713-747. 10.1512/iumj.1981.30.30055View ArticleGoogle Scholar - Koblitz N:
*p-adic Numbers, p-adic Analysis, and Zeta-functions*. New York: Springer-Verlag; 1984.View ArticleGoogle Scholar - Lapidus ML: Spectral and fractal geometry: From the Weyl–Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function. In
*Differential Equations and Mathematical Physics (Birmingham, 1990)*. Edited by: Bennewitz C. New York: Academic Press; 1992:151-182.View ArticleGoogle Scholar - Lapidus ML:
*In Search of the Riemann Zeros: Strings, fractal membranes and noncommutative spacetimes*. Providence, RI: Amer. Math. Soc; 2008.View ArticleGoogle Scholar - Lapidus ML, Hung L: Nonarchimedean Cantor set and string.
*J Fixed Point Theory Appl*2008, 3(1):181-190. 10.1007/s11784-008-0062-9View ArticleGoogle Scholar - Lapidus ML, Hung L: Self-similar p-adic fractal strings and their complex dimensions. p-adic numbers.
*Ultrametric Analysis and Applications*2009, 1(2):167-180.Google Scholar - Lapidus ML, Maier H: The Riemann Hypothesis and inverse spectral problems for fractal strings.
*J Lond Math Soc*1995, 2(52):15-34.View ArticleGoogle Scholar - Lapidus ML, Pearse EPJ: Tube formulas and complex dimensions of self-similar tilings.
*Acta Appl Math*2006, 112: 91-136.View ArticleGoogle Scholar - Lapidus ML, Pearse EPJ: Tube formulas for self-similar fractals. In
*Analysis on Graphs and Its Applications*. Providence, RI: Proc Symp Pure Math, Amer Math Soc; 2008:1-19.Google Scholar - Lapidus ML, Pomerance C: The Riemann zeta-function and the one-dimensional Weyl–Berry conjecture for fractal drums.
*Proc Lond Math Soc*1993, 3(66):41-69.View ArticleGoogle Scholar - Lapidus ML, van Frankenhuijsen M:
*Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions*. Boston: Birkhh¨auser; 2000.View ArticleGoogle Scholar - Lapidus ML, van Frankenhuijsen M:
*Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and spectra of fractal strings*. New York: Springer Monographs in Mathematics, Springer-Verlag; 2006.View ArticleGoogle Scholar - Lee JS: Periodicity on Cantor sets.
*Comm Korean Math Soc*1998, 13(3):595-601.Google Scholar - Hung L:
*p-adic Fractal Strings and Their Complex Dimensions, Ph.D. Dissertation*. Riverside: University of California; 2007.Google Scholar - Madore DA:
*A first introduction to p-adic numbers*. 2000. http://www.madore.org/~david/math/padics.pdfGoogle Scholar - Mendes P: Sum of Cantor sets: self-similarity and measure.
*Proc Amer Math Soc*1999, 127: 3305-3308. 10.1090/S0002-9939-99-05107-2View ArticleGoogle Scholar - Peitgen HO, Jürgens H, Saupe D:
*Chaos and Fractals: New Frontiers of Science*. 2nd edition. New York: Springer Verlag; 2004.View ArticleGoogle Scholar - Rani M, Prasad S: Superior Cantor sets and superior Devil’s staircases.
*Int J Artif Life Res*2010, 1(1):78-84.View ArticleGoogle Scholar - Robert AM:
*A Course in p-adic Analysis, Graduate Texts in Mathematics*. New York: Springer Verlag; 2000.View ArticleGoogle Scholar - Schikhof WH:
*Ultrametric calculus: An introduction to p-adic analysis, Cambridge Studies in Advanced Mathematics*. Cambridge: Cambridge Univ. Press; 1984.Google Scholar - Shaver C:
*An Exploration of the Cantor set, Mathematics Seminar*. 2010. http://www.rose-hulman.edu/mathjournal/archives/2010/vol11-n1/paper1/v11n1-1pd.pdfGoogle Scholar - Vladimirov VS, Volovich IV, Zelenov EI:
*p-Adic Analysis and Mathematical Physics*. Singapore: World Scientific Publ; 1994.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.