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

A sequence (*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

n=\frac{}{{s}_{k-1}\phantom{\rule{0.24em}{0ex}}{s}_{k-2}....\phantom{\rule{0.36em}{0ex}}{s}_{1}\phantom{\rule{0.12em}{0ex}}{s}_{0}\phantom{\rule{0.12em}{0ex}}}

is its *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:

\phantom{\rule{0.12em}{0ex}}\begin{array}{cc}\hfill {\mathbb{Z}}_{p}=\{{s}_{0}+{s}_{1}{p}^{1}+{s}_{2}{p}^{2}+\dots ;{s}_{i}\in \left(0,1,2,\dots ,p-1\right),\hfill & \hfill \mathrm{for}\phantom{\rule{0.12em}{0ex}}\mathrm{all}\phantom{\rule{0.37em}{0ex}}i\ge 0\}\hfill \end{array}

For this *p*-adic expansion, we can also write

{\mathbb{Z}}_{p}={\displaystyle \underset{c=0}{\overset{p-1}{\cup}}\left(c+p{\mathbb{Z}}_{p}\right),}

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.

Figure 1 below shows the representation of 5-adic 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.

N={\displaystyle \underset{k\in \mathbb{N}}{\cap}{N}_{k}}.

#### Theorem 3.1

Let *f*_{1}, *f*_{2} and *f*_{3} be the similarity contraction mappings on 5-adic integer *ℤ*_{5} defined by

\begin{array}{ccc}\hfill {f}_{1}\left(x\right)=5x,\hfill & \hfill {f}_{2}\left(x\right)=5x+2,\hfill & \hfill {f}_{3}\left(x\right)=5x+4,\hfill \end{array}

(1)

with scaling ratio 1/5. Then, the 5-adic Cantor one-fifth set *N* satisfies the self-referential equation

N={f}_{1}\left[N\right]\cup {f}_{2}\left[N\right]\cup {f}_{3}\left[N\right].

(2)

Proof: Using above construction of 5-adic Cantor one-fifth set, we can say that

{N}_{k+1}={f}_{1}\left[{N}_{k}\right]\cup {f}_{2}\left[{N}_{k}\right]\cup {f}_{3}\left[{N}_{k}\right]

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
}],

therefore, *f*_{1}[*N*] ∪ *f*_{2}[*N*] ∪ *f*_{3}[*N*] = ( ∩ *f*_{1}[*N*_{
k
}]) ∪ ( ∩ *f*_{2}[*N*_{
k
}]) ∪ ( ∩ *f*_{3}[*N*_{
k
}])

{f}_{1}\left[N\right]\cup {f}_{2}\left[N\right]\cup {f}_{3}\left[N\right]=\cap \left({f}_{1}\left[{N}_{k}\right]\cup {f}_{2}\left[{N}_{k}\right]\cup {f}_{3}\left[{N}_{k}\right]\right)

{f}_{1}\left[N\right]\cup {f}_{2}\left[N\right]\cup {f}_{3}\left[N\right]=\cap {N}_{k+1}=N

{f}_{1}\left[N\right]\cup {f}_{2}\left[N\right]\cup {f}_{3}\left[N\right]=N

which gives the proof of the theorem.

Figure 2 shows the graphical representation of 5-adic Cantor one-fifth set using iterated function system (*f*_{1}, *f*_{2}, *f*_{3}).

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

#### Theorem 3.2

The 5-adic Cantor one-fifth set is represented by the quinary expansion of its elements in the form

N=\left\{x\in {\mathbb{Z}}_{5}:x={x}_{0}+{5}^{1}{x}_{1}+{5}^{2}{x}_{2}+\dots ,{x}_{j}\in \left\{0,2,4\right\}\phantom{\rule{0.12em}{0ex}}\right\}

(3)

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

Proof: Let us define the inverse of similarity contraction mappings *f*_{1}, *f*_{2} and *f*_{3}, on *ℤ*_{5} as follows:

\begin{array}{ccc}\hfill {f}_{1}^{-1}\left(x\right)=x/5,\hfill & \hfill {f}_{2}^{-1}\left(x\right)=(x-2)/5,\hfill & \hfill {f}_{3}^{-1}\left(x\right)=(x-4)/5,\hfill \end{array}

(4)

Now, for *x*_{
j
} ∊ {0, 1, 2, 3, 4}, for all *j* ≥ 0, either

\phantom{\rule{0.12em}{0ex}}\begin{array}{ccc}\hfill x={x}_{0}+{5}^{1}{x}_{1}+{5}^{2}{x}_{2}+\dots ,\in 1+5{\mathbb{Z}}_{5}\hfill & \hfill \mathrm{or}\hfill & \hfill 3+5{\mathbb{Z}}_{5},\hfill \end{array}

(5)

if and only if 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

\begin{array}{l}{f}_{1}^{-1}\left(x\right)={f}_{2}^{-1}\left(x\right)={f}_{3}^{-1}\left(x\right)={x}_{1}+{5}^{1}{x}_{2}+\dots ,+{5}^{\eta -1}{x}_{\eta}+{5}^{\eta}{x}_{\eta +1}+\dots ,\hfill \\ {f}_{1}^{-1}\left(x\right)={f}_{2}^{-1}\left(x\right)={f}_{3}^{-1}\left(x\right)={x}_{1}+{5}^{1}{x}_{2}+\dots ,+{5}^{\mu -1}{x}_{\mu}+{5}^{\mu}{x}_{\mu +1}+\dots \hfill \end{array}

again repeating the process in this manner, we obtain the general case

\begin{array}{l}{f}_{1}^{-1}\left(x\right)={f}_{2}^{-1}\left(x\right)={f}_{3}^{-1}\left(x\right)={x}_{\eta}+5{x}_{\eta +1}+\dots ,\hfill \\ {f}_{1}^{-1}\left(x\right)={f}_{2}^{-1}\left(x\right)={f}_{3}^{-1}\left(x\right)={x}_{\mu}+5{x}_{\mu +1}+\dots \hfill \end{array}

which lie in the intervals 1 + 5*ℤ*_{5} and 3 + 5*ℤ*_{5} respectively. Thus, we found that

\begin{array}{ccc}\hfill N\cap \left(1+5{\mathbb{Z}}_{5}\right)=\varnothing \hfill & \hfill \mathrm{and}\hfill & \hfill N\cap \left(3+5{\mathbb{Z}}_{5}\right)=\varnothing \hfill \end{array}

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*.

Conversely, let all the variables *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,

x\notin \left\{\left({\displaystyle \underset{l=0}{\overset{\infty}{\cup}}{\displaystyle \underset{j\in {W}_{l}}{\cup}{f}_{j}}}\left(1+5{\mathbb{Z}}_{5}\right)\right)\cup \left({\displaystyle \underset{l=0}{\overset{\infty}{\cup}}{\displaystyle \underset{j\in {W}_{l}}{\cup}{f}_{j}}}\left(3+5{\mathbb{Z}}_{5}\right)\right)\right\}=Y

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

### Cantor one-fifth set as fractal string

It is well known from the definition of fractal string that such a set consists of countably many disjoint open intervals. The lengths of which form a sequence *L* = ℓ_{1}, ℓ_{2}, ℓ_{3}, …, called the lengths of the string. We can assume without loss of generality that

{\ell}_{1}\ge {\ell}_{2}\ge {\ell}_{3},\dots ,>0

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.

The Cantor one-fifth string ℵ, is the complement of [0, 1] of the usual Cantor one-fifth set *F*. The Figure 3 shows the geometrical representation of Cantor one-fifth string.

Thus, we obtain

\begin{array}{l}\aleph =\left(1/5,2/5\right)\cup (3/5,4/5)\cup (1/25,2/25)\cup (3/25,4/25)\cup \hfill \\ (11/25,12/25)\cup \left(1/25,2/25\right)\cup (13/25,14/25)\cup \hfill \\ (21/25,22/25)\cup (23/25,24/25)\cup p\hfill \end{array}

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, ....

Thus, the geometric zeta function of the Cantor one-fifth string is determined by the sequence ℵ:

\begin{array}{l}{\varsigma}_{\aleph}\left(s\right)={\displaystyle \sum _{k=0}^{\infty}{m}_{k}}{\ell}_{k}^{s}=\sum _{k=0}^{\infty}{2.3}^{k}{.5}^{\left(-k-1\right)s}=\frac{{2.5}^{s-1}}{{5}^{s}-3}\\ \phantom{\rule{6em}{0ex}}\mathrm{for}\phantom{\rule{2em}{0ex}}\mathrm{Re}\left(s\right)>log3/log5\end{array}

(6)

The poles of the such function are the set of complex numbers (see (Lapidus and Hung 2008), pp. 7) and given by

{D}_{L}=\left\{D+\mathit{inp}:n\in \mathbb{Z}\right\},=\left\{0.6826+\mathit{in}2\pi /log5:n\in \mathbb{Z}\right\},

(7)

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.

Further, representation of Cantor one-fifth string may be seen in Figure 4 using fractal harp.

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

Since, the construction of 5-adic Cantor one-fifth string (*ξ*) is analogue to the usual Cantor one-fifth set. We start, by subdividing the interval *ℤ*_{5} into closed subintervals

\begin{array}{c}\hfill {f}_{1}\left({\mathbb{Z}}_{5}\right)=0+5{\mathbb{Z}}_{5}\hfill \\ \hfill {f}_{2}\left({\mathbb{Z}}_{5}\right)=2+5{\mathbb{Z}}_{5}\hfill \\ \hfill {f}_{3}\left({\mathbb{Z}}_{5}\right)=4+5{\mathbb{Z}}_{5}\hfill \end{array}

since, fractal string is complement of the usual Cantor one-fifth set in the closed interval [0, 1], the remaining open subintervals after this step are given by

\begin{array}{l}{\mathbb{Z}}_{5}-{\displaystyle \underset{j=1}{\overset{2}{\cup}}{f}_{j}}\left({\mathbb{Z}}_{5}\right)=1+5{\mathbb{Z}}_{5}={G}_{1},\hfill \\ {\mathbb{Z}}_{5}-{\displaystyle \underset{j=2}{\overset{3}{\cup}}{f}_{j}}\left({\mathbb{Z}}_{5}\right)=3+5{\mathbb{Z}}_{5}={G}_{2}\hfill \end{array}

then, the *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:

{l}_{1}={l}_{2}={\mu}_{H}\left({G}_{1}\right)={\mu}_{H}\left({G}_{2}\right)=1/5

Again repeating the same process, by subdividing the closed intervals of first step (see Figure 1), we get

\begin{array}{l}{f}_{11}\left[{\mathbb{Z}}_{5}\right]=0+25{\mathbb{Z}}_{5},\phantom{\rule{2em}{0ex}}{f}_{21}\left[{\mathbb{Z}}_{5}\right]=10+25{\mathbb{Z}}_{5},\\ {f}_{31}\left[{\mathbb{Z}}_{5}\right]=20+25{\mathbb{Z}}_{5},\phantom{\rule{1.5em}{0ex}}{f}_{12}\left[{\mathbb{Z}}_{5}\right]=2+25{\mathbb{Z}}_{5},\\ {f}_{22}\left[{\mathbb{Z}}_{5}\right]=12+25{\mathbb{Z}}_{5},\phantom{\rule{1.5em}{0ex}}{f}_{32}\left[{\mathbb{Z}}_{5}\right]=22+25{\mathbb{Z}}_{5},\\ {f}_{13}\left[{\mathbb{Z}}_{5}\right]=4+25{\mathbb{Z}}_{5},\phantom{\rule{2em}{0ex}}{f}_{23}\left[{\mathbb{Z}}_{5}\right]=14+25{\mathbb{Z}}_{5},\\ {f}_{33}\left[{\mathbb{Z}}_{5}\right]=24+25{\mathbb{Z}}_{5}.\end{array}

Thus, the remaining open subintervals are given by

\begin{array}{ll}{\mathbb{Z}}_{5}-{\displaystyle \underset{j=1}{\overset{2}{\cup}}{f}_{j1}}\left({\mathbb{Z}}_{5}\right)=5+25{\mathbb{Z}}_{5}={G}_{3},\hfill & {\mathbb{Z}}_{5}-{\displaystyle \underset{j=2}{\overset{3}{\cup}}{f}_{j1}}\left({\mathbb{Z}}_{5}\right)=15+25{\mathbb{Z}}_{5}={G}_{4},\hfill \\ {\mathbb{Z}}_{5}-{\displaystyle \underset{j=1}{\overset{2}{\cup}}{f}_{j2}}\left({\mathbb{Z}}_{5}\right)=7+25{\mathbb{Z}}_{5}={G}_{5},\hfill & {\mathbb{Z}}_{5}-{\displaystyle \underset{j=2}{\overset{3}{\cup}}{f}_{j2}}\left({\mathbb{Z}}_{5}\right)=17+25{\mathbb{Z}}_{5}={G}_{6},\hfill \\ {\mathbb{Z}}_{5}-{\displaystyle \underset{j=1}{\overset{2}{\cup}}{f}_{j3}}\left({\mathbb{Z}}_{5}\right)=9+25{\mathbb{Z}}_{5}={G}_{7},\hfill & {\mathbb{Z}}_{5}-{\displaystyle \underset{j=2}{\overset{3}{\cup}}{f}_{j3}}\left({\mathbb{Z}}_{5}\right)=19+25{\mathbb{Z}}_{5}={G}_{8}.\hfill \end{array}

The subring *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

\begin{array}{l}{l}_{3}={l}_{4}={l}_{5}={l}_{6}={l}_{7}={l}_{8}={\mu}_{H}\left({G}_{3}\right)={\mu}_{H}\left({G}_{4}\right)\\ \phantom{\rule{1em}{0ex}}={\mu}_{H}\left({G}_{5}\right)={\mu}_{H}\left({G}_{6}\right)={\mu}_{H}\left({G}_{7}\right)={\mu}_{H}\left({G}_{8}\right)=1/25.\end{array}

Repeating the same process over and over again, we obtain a sequence ℓ_{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:

\begin{array}{l}\xi =\left(1+5{\mathbb{Z}}_{5}\right)\cup (3+5{\mathbb{Z}}_{5})\cup (5+25{\mathbb{Z}}_{5})\cup (15+25{\mathbb{Z}}_{5})\cup \hfill \\ \left(7+25{\mathbb{Z}}_{5}\right)\cup (17+25{\mathbb{Z}}_{5})\cup (9+25{\mathbb{Z}}_{5})\cup (19+25{\mathbb{Z}}_{5})\cup ....\hfill \end{array}

From Definition 2.3 (Lapidus and Hung 2009), the geometric zeta function of *ξ* is given by

\begin{array}{l}{\varsigma}_{\xi}={\left({\mu}_{H}\left(1+5{\mathbb{Z}}_{5}\right)\right)}^{s}+{\left({\mu}_{H}\left(3+5{\mathbb{Z}}_{5}\right)\right)}^{s}+{\left({\mu}_{H}\left(5+25{\mathbb{Z}}_{5}\right)\right)}^{s}+\dots \\ \phantom{\rule{1em}{0ex}}={\displaystyle \sum _{k=1}^{\infty}{m}_{k}}{\ell}_{k}^{s}={\displaystyle \sum _{k=1}^{\infty}{2.3}^{k}{.5}^{\left(-k-1\right)s}}=\frac{{2.5}^{s-1}}{{5}^{s}-3}\\ \phantom{\rule{7em}{0ex}}\mathrm{for}\phantom{\rule{1em}{0ex}}\mathrm{Re}\left(s\right)>log3/log5\end{array}

(8)

the poles of the such function are the set of complex numbers

{D}_{L}=\left\{D+\mathit{inp}:n\in \mathbb{Z}\right\}=\frac{log3}{log5}+\mathit{in}\frac{2\pi}{log5},

(9)

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*.