On one fractal property of the Minkowski function

TL;DR

This paper investigates a specific fractal property of the Minkowski function, demonstrating that it does not preserve the Hausdorff-Besicovitch dimension, which has implications for understanding its geometric behavior.

Contribution

The article reveals that the Minkowski function does not preserve the Hausdorff-Besicovitch dimension, providing new insights into its fractal properties.

Findings

Minkowski function does not preserve Hausdorff-Besicovitch dimension

Results presented at a scientific conference in 2011

The study contributes to fractal geometry understanding

Abstract

It is shown in the present article that the Minkowski function does not preserve the Hausdorff-Besicovitch dimension. Results of this article were presented by the author of this article on the Second Interuniversity Scientific Conference on Mathematics and Physics for Young Scientists in April, 2011 (Links to the conference paper (in Ukrainian): www.researchgate.net/publication/301637057, www.researchgate.net/publication/311665828. A link to the presentation (in Ukrainian): www.researchgate.net/publication/303752636).

Equations

$$G(x)=2^{1-a_1}-2^{1-(a_1+a_2)}+...+(-1{)}^{n-1}{2}^{1-(a_1+a_2+...+a_n)}+...,$$

$$[0;1]\ni x=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\dots }}}\equiv [0;a_1,a_2,...,a_n,...],a_n\in \mathbb{N}.$$

$$E_9\equiv \left\{x:x=[0;a_1,a_2,...,a_i,...],a_i\in \{1,2,3,...,9\}\forall i\in \mathbb{N}\right\}.$$

$$1-\frac{1}{n\lg 2}\le {\alpha }_0(E_n)\le 1-\frac{1}{8n\lg n},n>8.$$

$$0,6308969\le {\alpha }_0(E_9)\le 0,985445112.$$

$$\left\{y:y=G(x)=\frac{2}{2^{a_1}}-\frac{2}{2^{a_1+a_2}}+...+\frac{2(-1{)}^{n-1}}{2^{a_1+a_2+...+a_n}}+...,x=[0;a_1,a_2,...,a_n,...]\right\},$$

$${\left(\frac{1}{2^{k_1}}\right)}^{{\alpha }_0}+{\left(\frac{1}{2^{k_2}}\right)}^{{\alpha }_0}+{\left(\frac{1}{2^{k_3}}\right)}^{{\alpha }_0}+...+{\left(\frac{1}{2^{k_S}}\right)}^{{\alpha }_0}=1.$$

$$=\left\{y:y=\frac{2}{2^{a_1}}-\frac{2}{2^{a_1+a_2}}+...+\frac{2(-1{)}^{n-1}}{2^{a_1+a_2+...+a_n}}+...+\frac{2}{2^{a_1+a_2+...+a_n}}\left(\frac{(-1{)}^n}{2^{a_{n+1}}}+\frac{(-1{)}^{n+1}}{2^{a_{n+1}+a_{n+2}}}+...\right)\right\}$$

$${\Delta }_0\equiv \left\{y:y=G(x)=\frac{2}{2^{a_1}}-\frac{2}{2^{a_1+a_2}}+...+\frac{2(-1{)}^{n-1}}{2^{a_1+a_2+...+a_n}}+...\right\}.$$

$$D_n\equiv \left\{y:y=\frac{(-1{)}^n}{2^{a_{n+1}}}+\frac{(-1{)}^{n+1}}{2^{a_{n+1}+a_{n+2}}}+\frac{(-1{)}^{n+2}}{2^{a_{n+1}+a_{n+2}+a_{n+3}}}+...\right\}$$

$$\sup {\Delta }_0=\frac{2}{2^{k_1}}-\frac{2}{2^{k_1+k_2}}+\frac{2}{2^{k_1+k_2+k_1}}-\frac{2}{2^{k_1+k_2+k_1+k_2}}+...=$$

$$=\frac{\frac{2}{2^{k_1}}}{1-\frac{1}{2^{k_1+k_2}}}-\frac{\frac{2}{2^{k_1+k_2}}}{1-\frac{1}{2^{k_1+k_2}}}=\frac{2(2^{k_2}-1)}{2^{k_1+k_2}-1},$$

$$\inf {\Delta }_0=\{\frac{2}{2^{k_2}}-\frac{2}{2^{k_2+k_1}}+\frac{2}{2^{k_2+k_1+k_2}}-\frac{2}{2^{k_2+k_1+k_2+k_1}}+...=$$

$$=\frac{\frac{2}{2^{k_2}}}{1-\frac{1}{2^{k_1+k_2}}}-\frac{\frac{2}{2^{k_1+k_2}}}{1-\frac{1}{2^{k_1+k_2}}}=\frac{2(2^{k_1}-1)}{2^{k_1+k_2}-1}.$$

$$\lambda ({\Delta }_0)=\frac{2\left(2^{k_2}-2^{k_1}\right)}{2^{k_1+k_2}-1},$$

$${\Delta }_{k_{j_1}}=\left\{y:y=\frac{2}{2^{k_{j_1}}}-\frac{2}{2^{k_{j_1}+a_2}}+...+\frac{2(-1{)}^{n-1}}{2^{k_{j_1}+a_2+...+a_n}}+...\right\}$$

$${\Delta }_{k_1}=\left\{y:y=\frac{2}{2^{k_1}}\left(1-\frac{1}{2^{a_2}}+\frac{1}{2^{a_2+a_3}}-\frac{1}{2^{a_2+a_3+a_4}}+...\right)\right\}=\left\{\frac{2}{2^{k_1}}-\frac{2}{2^{k_1}}{D}_1\right\},$$

$${\Delta }_{k_2}=\left\{y:y=\frac{2}{2^{k_2}}\left(1-\frac{1}{2^{a_2}}+\frac{1}{2^{a_2+a_3}}-\frac{1}{2^{a_2+a_3+a_4}}+...\right)\right\}=\left\{\frac{2}{2^{k_2}}-\frac{2}{2^{k_2}}{D}_1\right\}.$$

$$\lambda ({\Delta }_{k_1})=\frac{2}{2^{k_1}}\lambda (D_1)=\frac{1}{2^{k_1}}\lambda ({\Delta }_0)$$

$$\lambda ({\Delta }_{k_2})=\frac{2}{2^{k_2}}\lambda (D_1)=\frac{1}{2^{k_2}}\lambda ({\Delta }_0).$$

$${\Delta }_{k_{j_1}{k}_{j_2}...k_{j_n}}\equiv \left\{y:y=\sum _{m=1}^n\frac{2(-1{)}^{m-1}}{2^{k_{j_1}+k_{j_2}+...+k_{j_m}}}+\sum _{t=n+1}^{\infty }\frac{2(-1{)}^{t-1}}{2^{k_{j_1}+k_{j_2}+...+k_{j_n}+a_{n+1}+...+a_t}}\right\},$$

$$\frac{\lambda \left({\Delta }_{k_{j_1}{k}_{j_2}...k_{j_{n-1}}{k}_{j_n}}\right)}{\lambda \left({\Delta }_{k_{j_1}{k}_{j_2}...k_{j_{n-1}}}\right)}=\frac{1}{2^{k_{j_n}}}\in \left\{\frac{1}{2^{k_1}},\frac{1}{2^{k_2}}\right\},$$

$$\frac{\lambda (D_{n+1})}{\lambda (D_n)}=1$$

$$D_{n+1}=D_{2l+1}\equiv \left\{y:y=-\frac{1}{2^{a_{2l+2}}}+\frac{1}{2^{a_{2l+2}+a_{2l+3}}}-\frac{1}{2^{a_{2l+2}+a_{2l+3}+a_{2l+4}}}+...\right\},$$

$$D_n=D_{2l}\equiv \left\{y:y=\frac{1}{2^{a_{2l+1}}}-\frac{1}{2^{a_{2l+1}+a_{2l+2}}}+\frac{1}{2^{a_{2l+1}+a_{2l+2}+a_{2l+3}}}-\frac{1}{2^{a_{2l+1}+a_{2l+2}+a_{2l+3}+a_{2l+4}}}+...\right\}.$$

$$\sup (D_{2l+1})=-\frac{\frac{1}{2^{k_2}}}{1-\frac{1}{2^{k_1+k_2}}}+\frac{\frac{1}{2^{k_1+k_2}}}{1-\frac{1}{2^{k_1+k_2}}}=\frac{1-2^{k_1}}{2^{k_1+k_2}-1},$$

$$\inf (D_{2l+1})=-\frac{\frac{1}{2^{k_1}}}{1-\frac{1}{2^{k_1+k_2}}}+\frac{\frac{1}{2^{k_1+k_2}}}{1-\frac{1}{2^{k_1+k_2}}}=\frac{1-2^{k_2}}{2^{k_1+k_2}-1},$$

$$\lambda (D_{2l+1})=\frac{2^{k_2}-2^{k_1}}{2^{k_1+k_2}-1}.$$

$$\sup (D_{2l})=\frac{\frac{1}{2^{k_1}}}{1-\frac{1}{2^{k_1+k_2}}}-\frac{\frac{1}{2^{k_1+k_2}}}{1-\frac{1}{2^{k_1+k_2}}}=\frac{2^{k_2}-1}{2^{k_1+k_2}-1},$$

$$\inf (D_{2l})=\frac{\frac{1}{2^{k_2}}}{1-\frac{1}{2^{k_1+k_2}}}-\frac{\frac{1}{2^{k_1+k_2}}}{1-\frac{1}{2^{k_1+k_2}}}=\frac{2^{k_1}-1}{2^{k_1+k_2}-1},$$

Full text

It is well-known that the main problem of the fractals theory is a problem of calculating of dimension. In particular, the dimension is the Hausdorff-Besicovitch dimension (a fractal dimension). But, since some classes of sets have complicated determination, a calculation of a value of the dimension is a difficult and labour-consuming problem for these sets. Therefore, to simplify of fractal dimension calculation, a problem of searching of auxiliary facilities appears. Transformations preserving the Hausdorff-Besicovitch dimension (DP-transformations) are such facilities. The transformations help to simplify of sets determinations and to investigate of belonging to the class of DP-tranformations of other transformations.

Monotone singular distribution functions as transformations of the segment $[0;1]$ are very interesting for studying of DP-transformations. The present article is devoted to considering of fractal properties of one example of such functions. In the article a preserving of the Hausdorff-Besicovitch dimension by the Minkowski function is investigated. Results of the present article were presented by the author of the article on Second Interuniversity Scientific Conference on Mathematics and Physics for Young Scientists in April, 2011 [6].

The following function

[TABLE]

is called the Minkowski function. An argument $x$ of the function determined in terms of representation of real numbers by continued fractions, i. e.,

[TABLE]

To establish of the one-to-one correspondence between rational numbers and quadratic irrationalities, the function was introduced by Minkowski in [5]. A difficulty of investigation of preserving the Hausdorff-Besicovitch dimension by the Minkowski function is caused by singularity and a complexity of the argument determination of the function.

Let us consider the following set

[TABLE]

”The geometry of continued fractions” does not have a property of ”classical self-similarity”. It is the reason of rather greater difficulties for obtaining exact results of fractal properties of continued fractions sets.

The main researches, in which fractal properies of some types of sets of continued fractions are studied, there are the articles of Jarnik [4] (the sets $E_{n}$, that representation of its elements by continued fractions contains symbols that do not exceed $n$), Good [1] (the sets of continued fractions, whose elements $a_{n}(x)$ quickly tend to infinity), Hirst [3] (the sets of such continued fractions, whose elements belong to infinite sequence of positive integers and increase indefinitely) and in [2] Hensley clarified estimations for Hausdorff-Besicovitch dimension of $E_{n}$ such that were obtained by Jarnik and Good:

[TABLE]

Whence, for $E_{9}$ we obtain that

[TABLE]

The answer to the main question of this article follows from the next statement about a value of the Hausdorff-Besicovitch dimension of the set $G(E_{9})$.

[TABLE]

where $\forall{i\in\mathbb{N}}:a_{i}\in\{{k_{1}},{k_{2}},...,{k_{S}}\}$ and $\{{k_{1}},{k_{2}},...,{k_{S}}\}$ is a fixed tuple of positive integers for a fixed positive integer number $S>1$, can be calculate by the formula:

[TABLE]

[TABLE]

the set

[TABLE]

Call the set

[TABLE]

by an ”indicative set” of rank $n$.

Let $S=2$ and $k_{1}<k_{2}$, then $\frac{2}{2^{k_{1}}}>\frac{2}{2^{k_{2}}}$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

So,

[TABLE]

where $\lambda(\cdot)$ is a diameter of a set.

Consider the following two sets

[TABLE]

where $j_{1}\in\{1,2\}$ and $\{k_{1},k_{2}\}\ni$ $k_{j_{1}}$ is a fixed for $\Delta_{k_{j_{1}}}$. That is

[TABLE]

[TABLE]

It is easy to see that

[TABLE]

and

[TABLE]

In the second step we obtain the following four sets: $\Delta_{k_{1}k_{1}},\Delta_{k_{1}k_{2}},\Delta_{k_{2}k_{1}},\Delta_{k_{2}k_{2}}$. In the $n$th step we shall have $2^{n}$ sets

[TABLE]

where $k_{j_{1}},k_{j_{2}},...,k_{j_{n}}$ is a fixed tuple of numbers from $\{k_{1},k_{2}\}$, and the following expression is true for all $n\in\mathbb{N}$.

[TABLE]

The last-mentioned fact follows from the next proposition.

[TABLE]

holds for all $n\in\mathbb{N}$.

Really, let $n=2l$, $l\in\mathbb{N}$. Then

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Similarly,

[TABLE]

[TABLE]

[TABLE]

So, $\lambda(D_{2l+1})=\lambda(D_{2l})$. A proof of the equality is analogical for the case of $n=2l+1$.

Let $I_{k_{j_{1}}k_{j_{2}}...k_{j_{n-1}}k_{j_{n}}}$ be a segment, whose endpoints coincide with endpoints of the corresponding set $\Delta_{k_{j_{1}}k_{j_{2}}...k_{j_{n-1}}k_{j_{n}}}$. Since $\Delta_{0}\subset I_{0}$ and $\Delta_{0}$ is a perfect set and

[TABLE]

[TABLE]

the theorem proved for the case of $S=2$.

Let $S>2$ and $k_{1}<k_{2}<...<k_{S}$. Similarly, in the $n$th step we shall have the $S^{n}$ sets $\Delta_{k_{j_{1}}k_{j_{2}}...k_{j_{n}}}$. It is easy to see that

[TABLE]

where $k_{j_{n}}\in\{k_{1},k_{2},...,k_{S}\},S\in\mathbb{N},S>1$.

Since the set $\Delta_{0}$ is a compact self-similar set of the space $\mathbb{R}^{1}$, the Hausdorff-Besicovitch dimension $\alpha_{0}(\Delta_{0})$ of the set $\Delta_{0}$ is calculating by the formula:

[TABLE]

The theorem is proved. ∎

So, for $k_{1}=1,k_{2}=2,...,k_{9}=9$ we obtain that

[TABLE]

From (1) and (2) it follows that $\alpha_{0}(E_{9})\neq\alpha_{0}(G(E_{9}))$. So, the Minkowski function does not preserve the Hausdorff-Besicovitch dimension.