Progress on Fractal Dimensions of the Weierstrass Function and Weierstrass-Type Functions

Abstract

The Weierstrass function $W\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{a}^{n}cos\left(2\pi b^{n}x\right)$ is a function that is continuous everywhere and differentiable nowhere. There are many investigations on fractal dimensions of the Weierstrass function, and the investigation of its Hausdorff dimension is still ongoing. In this paper, we summarize past researchers’ investigations on fractal dimensions of the Weierstrass function graph.

1. Introduction

Continuous functions are an important object of study in classical calculus, and one class of them is known as fractal functions. Fractal functions are an important part of fractal geometry, and their graphs are characterized by fractals. A class of classical fractal functions is known as Weierstrass functions.

The Weierstrass function was proposed in 1872. Prior to this, mathematicians believed that continuous functions were continuous everywhere except for a few special instances. The Weierstrass function changed mathematicians’ view of continuous functions at that time. Pathological functions and curves had been constructed under the influence of the Weierstrass function. They directly contributed to the creation of fractal geometry.

In this paper, we focus on the progress surrounding fractal dimensions of the Weierstrass function and Weierstrass-type functions, such as [1]

$$f\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{a}_{n}g(b_{n}x+{\theta }_n),$$

where g is a periodic Lipschitz function and ${\displaystyle \frac{a_{n+1}}{a_n}}\to 0,{\displaystyle \frac{b_{n+1}}{b_n}}\to \infty$$(n\to +\infty )$.

There have been many investigations on fractal dimensions’ estimation of the Weierstrass function, such as box dimension, Hausdorff dimension, K dimension and Packing dimension, but its Hausdorff dimension is still being researched. Due to the specificity of the Weierstrass function and the complexity of calculating the Hausdorff dimension, the lower bound has not been precisely calculated yet.

The study of Weierstrass functions can contribute to the development of mathematical analysis and the establishment of a rigorous system of real numbers. At the same time, the Weierstrass function has important applications in the description of turbulence phenomena, the simulation of stock curves, signal characterization and other fractal phenomena. Therefore, it is important to study it.

2. Notions

In this paper, unless otherwise stated, all subjects are entirely real.

2.1. Basic Notions

Let n be a positive integer and ${\mathbb{R}}^n$ be the n-dimensional Euclidean space. Let I be the unit closed interval $[0,1]$. Let $y=f\left(x\right)$ be defined in some neighborhood $U\left(x_0\right)$ of $x_0$. If there exists a limit of $f\left(x\right)$ when $x_0$ with

$${\displaystyle \underset{x\to x_0}{lim}}f\left(x\right)=f\left(x_0\right),$$

then $f\left(x\right)$ is said to be continuous at $x_0$ and the point $x_0$ is said to be the point of continuity of the function.

Let $C[a,b]$ be the set consisting of all continuous functions on the closed interval $[a,b]$ and define $d(f,g)={\displaystyle \underset{t\in [a,b]}{max}}|f\left(t\right)-g\left(t\right)|$, where $f\left(t\right),g\left(t\right)\in C[a,b]$. Then, $C[a,b]$ is said to be the space of continuous functions by distance d, denoted as $\left(C\right[a,b],d)$. A function space $C^k$ is the class of functions consisting of all continuously differentiable functions of order k on the same set in an Euclidean space.

Consider the function $f:[a,b]\to \mathbb{R}$.

$$graphf=\left\{\right(t,f\left(t\right)),a\le t\le b\}$$

is the graph of a subset of the coordinates of $(t,x)$ [2]. A function $f:X\to Y$ is called a Hölder function of index $\alpha ,\alpha \in [0,1]$ if there exists a constant $c\ge 0$, such that

$$|f\left(x\right)-f\left(y\right)|\le {c|x-y|}^{\alpha }(x,y\in X).$$

When $\alpha =1$, it is called a Lipschitz function, i.e.,

$$\left|f\right(x)-f(y\left)\right|\le c|x-y|(x,y\in X).$$

A probability space is a triad of the form $(\Omega ,F,\mathbb{P})$, where $(\Omega ,F)$ is a measurable space and $\mathbb{P}$ is a probability measure. Probability space satisfies the following three conditions:

• For any $A\in F$, there are $0\le P\left(A\right)\le 1;$
• $P(\Omega )=1;$
• If ${\left\{A_i\right\}}_{i=1}^{\infty }\subseteq F$, and different $A_i,A_j$ are disjointed, then $P\left({\displaystyle \underset{i=1}{\overset{\infty }{\cup }}}{A}_i\right)={\displaystyle \sum _{i=1}^{\infty }}P\left(A_i\right).$

2.2. Fractal Dimensions

There are several common definitions of fractal dimensions, such as box dimension, Hausdorff dimension, K dimension and Packing dimension, which can be denoted as ${dim}_H$, ${dim}_B$, ${dim}_P$ and ${dim}_K$, respectively.

Definition 1

([2]). Suppose that F is a bounded and non-empty subset of ${\mathbb{R}}^n$ and that s is a non-negative number. For any $\delta >0$, define

$${\mathcal{H}}_{\delta }^s\left(F\right)=inf\left\{{\displaystyle \sum _{i=1}^{\infty }}|{\mathrm{U}}_i{|}^s:\left\{{\mathrm{U}}_i\right\}\mathit{is}\mathit{a}\delta \text{-}\mathit{cover}\mathit{of}F\right\}.$$

As δ decreases, the class of permissible covers of F is reduced. Therefore, the infimum ${\mathcal{H}}_{\delta }^s\left(F\right)$ increases, and so it approaches a limit: $\delta \to 0^{+}$. Let

$${\mathcal{H}}^s\left(F\right)={\displaystyle \underset{\delta \to 0^{+}}{lim}}{\mathcal{H}}_{\delta }^s\left(F\right).$$

This limit exists for any subset F of ${\mathbb{R}}^n$, though the limitation value can be (and usually is) 0 or ∞. ${\mathcal{H}}_{\delta }^s\left(F\right)$ is called the s-dimensional Hausdorff measure of F.

Definition 2

([2]). There is a critical value of s at which ${\mathcal{H}}^s\left(F\right)$ ‘jumps’ from $+\infty$ to 0. This critical value is called the Hausdorff dimension of F, and it is written as ${dim}_{H}F$. It is defined for any set $F\subset {\mathbb{R}}^n$. Formally,

$${dim}_{H}F=inf\{s\ge 0:{\mathcal{H}}^s\left(F\right)=0\}=sup\{s\ge 0:{\mathcal{H}}^s\left(F\right)=\infty \}.$$

The box dimension is easier to compute than the Hausdorff dimension. It is covered with spheres of equal size. For many fairly regular sets, Hausdorff dimension and box dimension are equal. However, for most sets, the Hausdorff dimension is smaller than the box dimension.

Definition 3

([2]). Let F be any non-empty bounded subset of ${\mathbb{R}}^n$ and $N_{\delta }\left(F\right)$ be the smallest number of diameter sets at the highest δ, which can cover F. The lower and upper box dimensions of F, respectively, are defined as

$${{\displaystyle \underline{dim}}}_B\left(F\right)={\displaystyle \underset{\delta \to 0^{+}}{{\displaystyle \underline{lim}}}}{\displaystyle \frac{log{N}_{\delta }\left(F\right)}{-log\delta }}$$

and

$${\overline{dim}}_B\left(F\right)={\displaystyle \underset{\delta \to 0^{+}}{\overline{lim}}}{\displaystyle \frac{log{N}_{\delta }\left(F\right)}{-log\delta }}.$$

If these are equal, the common value is called the box dimension of F, as follows:

$${dim}_B\left(F\right)={\displaystyle \underset{\delta \to 0^{+}}{lim}}{\displaystyle \frac{log{N}_{\delta }\left(F\right)}{-log\delta }}.$$

The Hausdorff measure was introduced by using arbitrary open covers. Later on, there appeared a measure introduced in the form of a fill, defined as the Packing measure. This gave rise to the Packing dimension.

Definition 4

([2]). For $s\ge 0$ and $\delta >0$, let

$${\mathcal{P}}_{\delta }^s\left(F\right)=sup\left\{{\displaystyle \sum _{i=1}^{\infty }}|B_i{|}^s:\left\{B_i\right\}\mathit{is}\mathit{a}\mathit{collection}\mathit{of}\mathit{disjoint}\mathit{balls}\mathit{of}\mathit{radii}\mathit{at}\mathit{most}\mathit{\delta }\mathit{with}\mathit{centers}\mathit{in}\mathit{F}\right\}.$$

Since ${\mathcal{P}}_{\delta }^s\left(F\right)$ decreases with δ, the limit

$${\mathcal{P}}_0^s\left(F\right)={\displaystyle \underset{\delta \to 0^{+}}{lim}}{\mathcal{P}}_{\delta }^s\left(F\right)$$

exists.

Define ${\mathcal{P}}^s$ as the s-dimensional Packing measure using

$${\mathcal{P}}^s\left(F\right)=inf\left\{{\displaystyle \sum _i}{\mathcal{P}}_0^s\left(F_i\right):F\subset {\displaystyle \bigcup _{i=1}^{\infty }}{F}_i\right\}.$$

${\mathcal{P}}^s\left(F\right)$ is known as s-dimensional Packing measure. The Packing dimension is defined as follows:

$${dim}_{P}F=sup\{s:{\mathcal{P}}^s\left(F\right)=\infty \}=inf\{s:{\mathcal{P}}^s\left(F\right)=0\}.$$

Inspired by the Hausdorff dimension and box dimension, the K dimension was created, which combines these two fractal dimensions. Additionally, it is a natural tool used to solve Weierstrass-type functions.

Definition 5

([3]). Let f be a continuous function on I with $graphf$. For any open interval J of I, let $q_J$ be the least number of open squares $J\times J^{\prime }$, whose union covers $f\left(J\right)$. Denote $OSC(f,J)$ as the oscillation of f on J, i.e.,

$$OSC(f,J)={\displaystyle \underset{t,u\in J}{sup}}\left|f\right(t)-f(u\left)\right|.$$

It holds

$$q_J-1\le {\displaystyle \frac{OSC(f,J)}{\left|J\right|}}\le q_J.$$

Let $s\ge 0$ and P be the natural projection from ${\mathbb{R}}^2$ to $\mathbb{R}$, defined by $P(x,y)=x$. For any subset E of $graphf$ and arbitrary open cover $\mathcal{L}$ of $P\left(E\right)$, define

$${\phi }^s(E,\mathcal{L})={\displaystyle \sum _{J\in \mathcal{L}}}{q}_J{\left|J\right|}^s$$

and

$${\mathcal{K}}^s\left(E\right)={\displaystyle \underset{\delta \to 0^{+}}{lim}}{\displaystyle \underset{\left|\mathcal{L}\right|<\delta }{inf}}{\phi }^s(E,\mathcal{L})$$

where $\left|\mathcal{L}\right|=sup\left\{\right|J|:J\in \mathcal{L}\}.$

It can be proved that ${\mathcal{K}}^s\left(E\right)$ is an outer measure of $graphf$. Analogous to the Hausdorff dimension, the K dimension can be defined as

$${dim}_{K}graphf=inf\{s>0:{\mathcal{K}}^s\left(graphf\right)=0\}.$$

2.3. The Weierstrass Function

In 1872, Weierstrass [4] constructed a function using a number of term series

$$W\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{a}^{n}cos\left(2\pi b^{n}x\right)$$

where $a\in (0,1)$ and b is a positive odd number. Since then, the Weierstrass function has been extensively studied.

3. Progress on Fractal Dimensions of Weierstrass-Type Functions

In 1916, Hardy [5] proved that for $a\in (0,1),b>1,ab\ge 1,$

$$W\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{a}^{n}cos\left(2\pi b^{n}x\right)$$

is continuous everywhere and differentiable nowhere.

Theorem 1

([5]). Neither of the functions

$$C\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{a}^{n}cos\left(2\pi b^{n}x\right),S\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{a}^{n}sin\left(2\pi b^{n}x\right),$$

where $0<a<1,b>1,$ possesses a finite differential coefficient at any point in any case for

$$ab\ge 1.$$

In 1937, from a geometric point of view, Besicovitch and Ursell [6] looked the graph of the Weierstrass function as a fractal set and thus gave an estimation of the lower bound on the dimensionality of the function’s graph.

Theorem 2

([6]). The fractal dimension number d of the $graphf$, where $f\left(x\right)$ satisfies the Hölder condition with exponent δ, allows us to see that the inequalities $1\le d\le 2-\delta$ hold.

In 1977, Mandelbrot [7] offered the following conjecture: for the function

$$W\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{a}^{n}cos\left(2\pi b^{n}x\right),$$

the Hausdorff dimension of $graphW$ is equal to its box dimension when $b>2$.

Conjecture 1

([7]). For the Weierstrass function, Hardy proved that its box dimension is $2-\alpha$. The validity of the natural conjecture that the Hausdorff dimension is $2-\alpha$ needs to be checked.

In 1984, Kaplan, Mallet and Yorke [8] considered the graph of the function

$$f_{\lambda ,b}^{\varphi }\left(t\right)={\displaystyle \sum _{n=1}^{\infty }}{\lambda }^n\varphi \left(b^{n}t\right)$$

as an attractor of a certain dynamical system, where $\varphi$ is $\mathbb{Z}$-periodic and ${\displaystyle \frac{1}{b}}\le \lambda <1$. They proved that the box dimension of $graph{f}_{\lambda ,b}^{\varphi }$ is $2+{\displaystyle \frac{log\lambda }{logb}}$. For the general function

$$f\left(t\right)={\displaystyle \sum _{n=1}^{\infty }}{\lambda }^n\varphi (b^{n}t+{\theta }_n),$$

the corresponding conclusion also holds.

Theorem 3

([8]). Let $\alpha ={\displaystyle \frac{log(1/\lambda )}{logb}}$. Note that ${\varphi }_a$ is defined for all $a\in \mathbb{R}$,

$$\varphi \left(t\right)\sim {\displaystyle \sum _{\alpha }}{\varphi }_a{e}^{iat},$$

$${\varphi }_a={\displaystyle \underset{T\to +\infty }{lim}}{\displaystyle \frac{1}{2T}}{\int }_{-T}^T\varphi \left(t\right)e^{-iat}dt.$$

Define the doubly infinite formal sum $g\left(t\right)$ by

$$g\left(t\right)={\displaystyle \sum _{n=-\infty }^{\infty }}{\lambda }^n\varphi \left(b^{n}t\right).$$

The Fourier series of $g\left(t\right)$ is ${\displaystyle \sum _{\sigma }}{g}_{\sigma }{e}^{i\sigma t}$, where $g_{\sigma }={\displaystyle \sum _{k=-\infty }^{\infty }}{\lambda }^k{\varphi }_{\sigma b^{-k}}$.

Assume ${\displaystyle \sum _a}|{\varphi }_a{\left|\right|a|}^{\alpha }<\infty$ and $g_{{\sigma }_0}\ne 0$. Also assume that ${\varphi }^{\prime }\left(t\right)$ is uniformly bounded. Then,

$${dim}_{B}graphf=2-\alpha =2-\left|{\displaystyle \frac{log\lambda }{logb}}\right|.$$

In 1986, Mauldin and Williams [9,10] considered the function

$$W_b\left(x\right)={\displaystyle \sum _{n=-\infty }^{\infty }}{b}^{-\alpha n}[\Phi (b^{n}x+{\theta }_n)-\Phi \left({\theta }_n\right)]$$

where $b>1,0<\alpha <1$, each ${\theta }_n$ is an arbitrary number, and $\Phi$ has one period. They showed that there is a positive constant C, such that if b is large enough, ${dim}_{H}graph{W}_b\ge 2-\alpha -{\displaystyle \frac{C}{lnb}}.$

Theorem 4

([10]). If g is a Lipschitz function, then

$${dim}_{H}graph(f+g)={dim}_{H}graphf.$$

In particular, ${dim}_{H}graph{W}_b={dim}_{H}graph{f}_b$, where

$$f_b={\displaystyle \sum _{n=0}^{\infty }}{b}^{-\alpha n}\varphi (b^{n}x+{\theta }_n)$$

whenever ϕ is bounded and Lipschitz, each ${\theta }_n$ is an arbitrary number.

Similarly, in 2023, for fractal dimensions of the linear combination of continuous functions, Yu and Liang [11,12] provided the following conclusion:

Theorem 5

([11]). Let $F_s$ be the set of s-dimensional continuous functions on I. Let $f\left(x\right)\in F_{s_1}$, $g\left(x\right)\in F_{s_2}$ and $s_1\ne s_2$. Then,

$${dim}_{B}graph(f+g,I)=max\left\{{dim}_{B}graph(f,I),{dim}_{B}graph(g,I)\right\}.$$

Theorem 6

([10]). Suppose that $\Phi :\mathbb{R}\to \mathbb{R}$ satisfies the following:

1. 

Is non-constant and continuous;

2. 

Has a piece-wise continuous derivative;

3. 

$\Phi (x+1)=\Phi \left(x\right);$

4 

$\parallel \Phi \parallel =1.$

Fix $0<\alpha <1$. Let J be a subinterval of I with length $l>0$, such that

1. 

${\Phi }^{\prime }$ is continuous on $J;$

2. 

$inf{\Phi }^{\prime }\ge \epsilon >0$ on J.

There is a positive constant C, such that if $b\ge 3/l$, then

$${dim}_H{f}_b\ge 2-\alpha -{\displaystyle \frac{C}{lnb}}$$

where

$$f_b\left(x\right)={\displaystyle \sum _{n=-\infty }^{\infty }}{b}^{-\alpha n}\Phi (b^{n}x+{\theta }_n)$$

and ${\theta }_0,{\theta }_1,\dots ,{\theta }_n,\dots$ are arbitrary phases.

In 1989, Przytychi and Urbanski [13] considered the functions

$$f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{\varphi \left(b_{n}x\right)}{b_n^{\delta }}}$$

where $\varphi$ is a Lipschitz function with period 1 and the series ${\left\{b_n\right\}}_{n=1}^{\infty }$ satisfies certain condition. They proved that ${dim}_{H}graphf\ge 1$.

Theorem 7

([13]). If all the eigenvalues of $\tilde{f}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ are distinct, every pre-invariant non-trivial continuous $K\subset T^k$ (a compact connected set containing more than one point) has a Hausdorff dimension that is greater than 1.

In 1992, Ledrappier [14] provided a sufficient condition in order to determine the Hausdorff dimension of the graph. They computed the Hausdorff dimension of certain subsets of ${\mathbb{R}}^2$ and applied their methods in an example function as

$${\psi }_{s,\varphi }\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{2}^{n(s-2)}\varphi \left(2^{n}x\right)$$

where $\varphi :\mathbb{R}\to \mathbb{R}$ is a $C^0$, $\mathbb{Z}$-periodic Lipschitz function and $s\in [1,2)$. The measure $m_{s,\varphi }$ on $graph(\varphi ,s)$ projects onto the Lebesgue measure on $(0,1]$. The function ${\varphi }_0$ is denoted as the distance to the closest integer.

A number $\theta$ ($0<\theta <1$) is called an Erdös number if the distribution law of ${\displaystyle \sum _{n=0}^{\infty }}{\theta }_n{ϵ}_n$ has a dimension of 1, where ${ϵ}_n$ are Bernoulli variables with values of $+1$ or $-1$.

Theorem 8

([14]). Let $2^{1-s}$ be an Erdös number; then, ${dim}_H{m}_{s,{\varphi }_0}=s$.

Corollary 1

([14]). Let $2^{1-s}$ be an Erdös number; then, ${dim}_{H}graph(s,\varphi )=s$.

In 1993, Hu and Lau [3] proved that the K dimension is closer to the Hausdorff dimension than the other fractal dimensions for the Weierstrass-type function, as follows:

$$W\left(x\right)={\displaystyle \sum _{i=1}^{\infty }}{\lambda }^{-\alpha i}g\left({\lambda }^{i}x\right)$$

where $\lambda >1,0<\alpha <1$, and g is an almost periodic Lipschitz function of an order greater than $\alpha$. It has been shown that the K dimension of $graphW$ is equal to $2-\alpha$. This conclusion is also equivalent to a certain rate of the function’s local oscillation.

Let

$$V\left(x\right)={\displaystyle \sum _{n=-\infty }^{\infty }}{\lambda }^{-\alpha i}g\left({\lambda }^{i}x\right)$$

where $x\in \mathbb{R}$. The series converges for all x. V is known as the Weierstrass–Mandelbrot function.

Theorem 9

([3]). The following statements are equivalent:

1. 

$V\ne 0;$

2. 

There exist positive numbers ε, η and c, such that $|E(W,\epsilon ,I_x)|/|I_x|\ge c$ for every $x\in \mathbb{R}$ and every interval $I_x$ containing x with $|I_x|<\eta ;$

3. 

$0<K^{2-\alpha }\left(graphW\right)<\infty$, in this case ${dim}_{K}graphW=2-\alpha$.

Moreover, the above equivalence is still valid if W in (2) and (3) is replaced by V.

In 1998, for the function

$$W_{\Theta }\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{a}^{n}cos\left(2\pi (b^{n}x+{\theta }_n)\right)$$

where the ranges of the parameters $a,b$ are given and each ${\theta }_n$ is chosen independently with respect to the uniform probability measure in I, Hunt [15] proved that the Hausdorff dimension of $graph{W}_{\Theta }$ is $2+{\displaystyle \frac{loga}{logb}}$.

Theorem 10

([15]). If each ${\theta }_n(n\in {\mathbb{N}}^{+})$ is chosen independently with respect to the uniform probability measure in I, then, with a probability of one, the Hausdorff dimension of the graph of $W_{\Theta }$ is $D=2+{\displaystyle \frac{loga}{logb}}$.

In 2000, Wen [16] calculated the Hausdorff and box dimensions of the Besicovich function:

$$B\left(t\right)={\displaystyle \sum _{n=1}^{\infty }}{\lambda }_n^{s-2}cos\left({\lambda }_{n}t\right)$$

where $1<s<2,{\lambda }_n\to +\infty$ when $n\to +\infty$. He proved that

$${dim}_{H}graphB=1+{\displaystyle \underset{n\to +\infty }{liminf}}{\displaystyle \frac{(s-1)log{\lambda }_n}{(s-1)log{\lambda }_n+(2-s)log{\lambda }_{n+1}}}.$$

Theorem 11

([16]). Let ${\displaystyle \frac{{\lambda }_{n+1}}{{\lambda }_n}}\nearrow \infty (n\to +\infty )$; then,

$${dim}_{H}graphB=1+{\displaystyle \underset{n\to +\infty }{liminf}}{\displaystyle \frac{(s-1)log{\lambda }_n}{(s-1)log{\lambda }_n+(2-s)log{\lambda }_{n+1}}}.$$

In 2004, Yao, Su and Zhou [17] investigated the fractional-order differential and integral of the Weierstrass-type function:

$$W\left(t\right)={\displaystyle \sum _{j=1}^{\infty }}{\lambda }^{-\alpha j}sin\left({\lambda }^{j}t\right)$$

where $0<\alpha <1,\lambda >1.$ They provided an estimation of the K dimension of W on I.

Theorem 12

([17]). Let g be a μth-order fractional-order differential function of W. Let $0<\mu <\alpha <1$. When λ is sufficiently large,

$${dim}_{K}graph(g,I)=2-\alpha +\mu .$$

In 2011, for a function of the form

$$f\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{\varphi \left(b^{n}x\right)}{b_n^{\alpha }}}$$

where $\varphi$ is a smooth function and the sequence ${\left\{b_n\right\}}_{n=1}^{\infty }$ satisfies that there exist $B\ge 1,\mu \ge 0$—such that $b_{n+1}\ge B{b}_n,b_n\ge \mu b_{n+1}$ when B is sufficiently large—Biacino [18] proved that the Hausdorff dimension of $graphf$ is $2-\alpha$.

Theorem 13

([18]). Let

$$f\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{\varphi \left(b^{n}x\right)}{b_n^{\alpha }}}$$

where $0<\alpha <1$, ϕ is a smooth function and where ${\left\{b_n\right\}}_{n=1}^{\infty }$ is such that there exist two numbers $B>1,B\in \mathbb{N}$ and $\mu >0$, for which $b_{n+1}\ge B{b}_n$ and $b_n\ge \mu b_{n+1}$ for every $n\in {\mathbb{N}}^{+}$. Then, if B is large enough, the Hausdorff dimension of $graphf$ is maximum and equals to $2-\alpha$.

In 2011, Baranski [1] determined the Hausdorff and box dimensions of graphs for a general class of Weierstrass-type functions of the form

$$f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{a}_{n}g(b_{n}x+{\theta }_n)$$

where g is a periodic Lipschitz real function and ${\displaystyle \frac{a_{n+1}}{a_n}}\to 0,{\displaystyle \frac{b_{n+1}}{b_n}}\to +\infty$ as $n\to +\infty$. Moreover, for any $1\le H\le B\le 2$, they provided examples of such functions with ${dim}_{H}graphf={dim}_{B}graphf=H,{\overline{dim}}_{B}graphf=B$.

Theorem 14

([1]). Let $g:\mathbb{R}\to \mathbb{R}$ be a periodic Lipschitz function, such that g is strictly monotone on some (non-trivial) interval $I\subset \mathbb{R}$ with $\left|g\right(x)-g(y\left)\right|>\delta |x-y|$ for every $x,y\in I$ and a constant $\delta >0$. If $a_n$, $b_n>0$, ${\displaystyle \frac{a_{n+1}}{a_n}}\to 0$, ${\displaystyle \frac{b_{n+1}}{b_n}}\to +\infty$$(n\to +\infty )$ and ${\theta }_n\in \mathbb{R}$, then for functions f,

$${dim}_{H}graphf={{\displaystyle \underline{dim}}}_{B}graphf=1+{\displaystyle \underset{n\to \infty }{lim}}inf{\displaystyle \frac{{log}^{+}{d}_n}{log{b}_{n+1}{d}_n/d_{n+1}}},$$

$${\overline{dim}}_{B}graphf=1+{\displaystyle \underset{n\to \infty }{lim}}sup{\displaystyle \frac{{log}^{+}{d}_n}{log{b}_n}},$$

where $d_n={\sum }_{n=1}^{\infty }{a}_n{b}_n$ and ${log}^{+}$ is a positive function of logarithmic function.

Corollary 2

([1]). For every $1\le H\le B\le 2$, there exists a function f fulfilling the assumptions of Theorem 14, such that

$${dim}_{H}graphf={{\displaystyle \underline{dim}}}_{B}graphf=H,{\overline{dim}}_{B}graphf=B.$$

In 2014, Romanowska [19] considered functions of the type

$$f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{a}_{n}g(b_{n}x+{\theta }_n)$$

where $a_n$ denote independent random variables uniformly distributed on $(-a_n,a_n)$. Moreover, for some $0<a<1,{\displaystyle \frac{b_{n+1}}{b_n}}\ge b>1,a^{2}b>1$ and g is $C^1$, a periodic real function with a finite number of critical points in every bounded interval. They proved that the occupation measure for f has a density that is almost surely $L^2$. Furthermore, the Hausdorff dimension of $graphf$ is almost surely equal to $D=2+{\displaystyle \frac{loga}{logb}}$, provided $b={\displaystyle \underset{n\to +\infty }{lim}}{\displaystyle \frac{b_{n+1}}{b_n}}>1$ and $ab>1$.

Theorem 15

([19]). Assume that $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{a}_{n}g(b_{n}x+{\theta }_n)$ satisfies the following conditions:

1. 

${\left\{a_n\right\}}_{n=0}^{\infty }$ is a sequence of real independent random variables defined on some probabilistic space $(\Omega ,\mathbb{P})$ with a uniform distribution on $(-a^n,a^n)$ for $0<a<1$;

2. 

${\displaystyle \underset{n\to \infty }{lim}}{\displaystyle \frac{b_{n+1}}{b_n}}=b$ for $b>1,ab>1$;

3. 

${\theta }_n\in \mathbb{R}$ for $n\in {\mathbb{N}}^{+}$;

4. 

$g:\mathbb{R}\to \mathbb{R}$ is $C^1$ periodic and has a finite number of critical points in every bounded interval.

Then, the Hausdorff and box dimensions of $graphf$ are equal to

$${dim}_{H}graphf=D=2+{\displaystyle \frac{loga}{logb}}，$$

almost surely.

In 2014, Barański, Bárány and Romanowska [20] proved that when $b\ge 2$ and $\lambda$ is sufficiently close to 1,

$${dim}_{H}graphf=2+{\displaystyle \frac{loga}{logb}},$$

where $\varphi \in C^3$ and $\varphi$ is $\mathbb{Z}$-periodic.

Let

$$W_{\lambda ,b}\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{\lambda }^{n}cos\left(2\pi b^{n}x\right),$$

where $b\ge 2$ is an integer. It has been proved that for every b, there exists a ${\lambda }_b\in (1/b,1)$, such that the Hausdorff dimension of $graph{W}_{\lambda ,b}$ is equal to $D=2+{\displaystyle \frac{log\lambda }{loga}}$.

Let

$${\mu }_{\lambda ,b}^{\varphi }={(Id,f_{\lambda ,b}^{\varphi })}_{*}{\mathcal{L}|}_{[0,1]}$$

be the lift of the Lebesgue measure $\mathcal{L}$ on I with regard to the graph of $f_{\lambda ,b}^{\varphi }$.

The subscript ∗ denotes the push forward method, which is a method that is used for transferring measures from one space to another. The definition describes how the function transfers the Lebesgue measure from the set on the real line [0,1] to the set on the function’s graph.

It is well known that ${dim}_{B}graph{f}_{\lambda ,b}^{\varphi }\le D$, and it is sufficient to prove $dim{\mu }_{\lambda ,b}^{\varphi }=D$.

Theorem 16

([20]). For every integer $b\ge 2$,

$$dim{\mu }_{\lambda ,b}=2+{\displaystyle \frac{log\lambda }{logb}}$$

for every $\lambda \in ({\lambda }_b,1)$, where ${\lambda }_b$ is equal to the unique zero of the function

$$h_b\left(\lambda \right)=\left\{\begin{array}{ccc}{\displaystyle \frac{1}{4{\lambda }^2{(2\lambda -1)}^2}}+{\displaystyle \frac{1}{16{\lambda }^2{(4\lambda -1)}^2}}-{\displaystyle \frac{1}{8{\lambda }^2}}+{\displaystyle \frac{\sqrt{2}}{2\lambda }}-1,\hfill & & forb=2,\hfill \\ {\displaystyle \frac{1}{{(b\lambda -1)}^2}}+{\displaystyle \frac{1}{{(b^2\lambda -1)}^2}}-{sin}^2\left({\displaystyle \frac{\pi }{b}}\right),\hfill & & forb\ge 3\hfill \end{array}\right.$$

on $(1/b,1)$. In particular,

$${dim}_{H}graph{W}_{\lambda ,b}={dim}_{B}graph{W}_{\lambda ,b}=D$$

for every $\lambda \in ({\lambda }_b,1)$.

In 2015, building on the work of Ledrappier and others, Keller [21] provided an elementary proof that for all $\lambda \in ({\lambda }_b,1)$ with a suitable ${\lambda }_b<1$,

$${dim}_{H}graph(W_{\lambda ,b},I)=2+{\displaystyle \frac{loga}{logb}}.$$

This reproduces results without using the dimension theory for hyperbolic measures of Ledrappier and Young, being replaced by a simple telescoping argument together with a recursive multi-scale estimate.

Theorem 17

([21]). Let $\gamma ={\displaystyle \frac{1}{b\lambda }}<1$. For each integer $b\ge 2$, there exists a ${\lambda }_b<1$, such that $graph{W}_{\lambda ,b}$ has a Hausdorff dimension of $D=2+{\displaystyle \frac{log\lambda }{logb}}$ for every $\lambda \in ({\lambda }_b,1)$.

In 2015, Baranski [22] considered continuous functions of the form

$$f_{\lambda ,b}^{\varphi }\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{\lambda }^n\varphi \left(b^{n}x\right)$$

for $x\in \mathbb{R}$, where $b>1,{\displaystyle \frac{1}{b}}<\lambda <1$, and $\varphi :\mathbb{R}\to \mathbb{R}$ is a non-constant, $\mathbb{Z}$-periodic, Lipschitz continuous, piece-wise $C^1$ function. In that paper, they presented a survey of results on dimensions of Weierstrass-type function graphs on $\mathbb{R}$.

Theorem 18

([22]). Note that ${\mu }_{\lambda ,b}$ is the lift of the Lebesgue measure on I to graph $W_{\lambda ,b}$. For every integer $b>1$, there exist ${\lambda }_b,{\tilde{\lambda }}_b\in (1/b,1)$, such that for every $\lambda \in ({\lambda }_b,1)$ and Lebesgue, almost every $\lambda \in ({\tilde{\lambda }}_b,1)$ and $dim{\mu }_{\lambda ,b}=D$ (given in Theorem 16). In particular,

$${dim}_{H}graph{W}_{\lambda ,b}=D$$

for every $\lambda \in ({\lambda }_b,1)$ and almost every $\lambda \in ({\tilde{\lambda }}_b,1)$.

In 2018, Shen [23] proved that the Hausdorff dimension of the graph of the function ${\displaystyle \sum _{n=1}^{\infty }}{\lambda }^{n}cos\left(2\pi b^{n}x\right)$ is $2+{\displaystyle \frac{log\lambda }{logb}}$ for any integer $b\ge 2$ and ${\displaystyle \frac{1}{b}}<\lambda <1$.

Theorem 19

([23]). For any integer $b\ge 2$ and any $\lambda \in (b^{-1},1)$, the Hausdorff dimension of $graph{W}_{\lambda ,b}$ is equal to D (given in Theorem 16).

David [24] build a Laplacian on $graphW$. He came across a simpler means of computing box dimension of the graph of

$$W\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{\lambda }_{n}cos\left(2\pi N_b^{n}x\right)$$

where $\lambda$ and $N_b$ are two real numbers, such that $0<\lambda <1$, $N_b\in \mathbb{N}$ and $\lambda N_b>1$. They used a sequence graph that could approximate the studied one, bypassing all the aforementioned tools. The box dimension of the corresponding graph is exactly $D=2+{\displaystyle \frac{log\lambda }{log{N}_b}}$.

Corollary 3

([24]). The box dimension of $graphW$ is exactly $D=2+{\displaystyle \frac{log\lambda }{log{N}_b}}$.

In 2021, for a real analytic periodic function $\varphi :\mathbb{R}\to \mathbb{R}$, an integer $b\ge 2$, and $\lambda \in \left({\displaystyle \frac{1}{b}},1\right)$, Ren [25] proved the following Dichotomy for the Weierstrass-type function

$$W\left(x\right)={\displaystyle \sum _{n\ge 0}}{\lambda }^n\varphi \left(b^{n}x\right).$$

Either $W\left(x\right)$ is real analytic or ${dim}_{H}graphW=2+{log}_b\lambda$. Furthermore, given b and $\varphi$, the former alternative only happens for finitely many $\lambda$, unless $\varphi$ is a constant.

Theorem 20

([25]). Let $b\ge 2$ be an integer, $\lambda \in (1/b,1)$, and ϕ be a $\mathbb{Z}$-peridoc, real analytic function. Then, exactly one of the following holds:

1. 

$W\left(x\right)={\displaystyle \sum _{n\ge 0}}{\lambda }^n\varphi \left(b^{n}x\right)$ is real analytic;

2. 

$graphW$ has a Hausdorff dimension

$$D=2+{log}_b\lambda .$$

Moreover, given b and non-constant ϕ, the first alternative only holds for finitely many $\lambda \in (1/b,1)$.

4. Conclusions

Starting from the Weierstrass function to Weierstrass-type functions, the investigation of fractal dimensions is still ongoing.

The box dimension of Weierstrass-type functions has been proved to be $2-\alpha$. Moreover, in certain conditions, the Packing dimension of Weierstrass-type functions is equal to their box dimensions. Scholars also studied the Hausdorff dimension of the Weierstrass function. Since the Hausdorff dimension is difficult to calculate, the problem is still in the process of being solved. During the calculation of the Hausdorff dimension, the K dimension of Weierstrass-type functions was introduced, which combines box and Hausdorff dimensions. It was proved that the K dimension is close to the Hausdorff dimension. When functions satisfy certain conditions, the K dimension of Weierstrass-type functions is $2-\alpha$. An attempt was made to prove a conjecture that the Hausdorff dimension is equal to the box dimension when $b>2$. For more recent work on the dimensional theory of Weierstrass-type functions, readers can consult [26,27,28,29], for instance.