Arithmetic on self-similar sets

TL;DR

This paper establishes conditions under which the continuous image of the product of two self-similar sets contains interior points, extending previous results by removing the need for a thickness product condition, with applications to q-expansions.

Contribution

It provides a new sufficient condition for the image of two self-similar sets under a continuous function to have interior, generalizing prior results by relaxing the thickness requirement.

Findings

The image contains interior points under the new condition.

The result does not require the product of thicknesses to exceed one.

Application to univoque sets in q-expansions.

Abstract

Let $K_1$ and $K_2$ be two one-dimensional homogeneous self-similar sets. Let $f$ be a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the continuous image of $f$ by $$ f_{U}(K_1,K_2)=\{f(x,y):(x,y)\in (K_1\times K_2)\cap U\}. $$ In this paper we give an sufficient condition which guarantees that $f_{U}(K_1,K_2)$ contains some interiors. Our result is different from Simon and Taylor's \cite[Proposition 2.9]{ST} as we do not need the condition that the multiplication of the thickness of $K_1$ and $K_2$ is strictly greater than $1$. As a consequence, we give an application to the univoque sets in the setting of $q$-expansions.

Equations

$$f_U(K_1,K_2)=\{f(x,y):(x,y)\in (K_1\times K_2)\cap U\}.$$

$$C+C=[0,2],C-C=[-1,1].$$

$$C÷C={\cup }_{-\infty }^{+\infty }{3}^n[2/3,3/2].$$

$$f_U(E,F)=\{f(x,y):(x,y)\in (E\times F)\cap U\}.$$

$$f_i(a)\le f_{i+1}(a),1\le i\le n-1,g_j(c)\le g_{j+1}(c),1\le j\le m-1.$$

$$D_1=\{1\le i\le n-1:f_i(b)-f_{i+1}(a)<0\},{\kappa }_1=\underset{i\in D_1}{\max }\{f_{i+1}(a)-f_i(b)\}.$$

$$D_2=\{1\le j\le m-1:g_j(d)-g_{j+1}(c)<0\},{\kappa }_2=\underset{j\in D_2}{\max }\{g_{j+1}(c)-g_j(d)\}.$$

$$\frac{{\kappa }_1}{d-c}<\left|\frac{{\partial }_{y}f{|}_{(x_0,y_0)}}{{\partial }_{x}f{|}_{(x_0,y_0)}}\right|<\frac{\lambda (b-a)}{{\kappa }_2},$$

$$\left\{\begin{array}[]{cc}\lambda(b-a)>\kappa_{2}\\ \kappa_{1}<d-c,\\ \end{array}\right.$$

$$x=\sum _{n=1}^{\infty }\frac{a_n}{q^n}.$$

$$U_q+U_q=\left[0,\frac{2}{q-1}\right],$$

$$x^3-x^2-2x+1=0.$$

$$1-2\lambda <\left|\frac{{\partial }_{y}f{|}_{(x_0,y_0)}}{{\partial }_{x}f{|}_{(x_0,y_0)}}\right|<\frac{\lambda }{1-2\lambda },$$

$$\left\{g_1(x)=\frac{x+1}{q^2},g_2(x)=\frac{x}{q^2}+\frac{1}{q}\right\},$$

$$U_q^2+U_q^2,U_q^2-U_q^2,U_q\cdot U_q,U_q÷U_q$$

$${\dim }_H(U_q\cdot U_q)={\dim }_H(U_q÷U_q)={\dim }_H(U_q^2\pm U_q^2)=\min \{2{\dim }_H(U_q),1\}.$$

$$\frac{\log q_1}{\log q_2}\notin \mathbb{Q},$$

$${\dim }_H(U_{q_1}+U_{q_2})=\min \{{\dim }_H(U_{q_1})+{\dim }_H(U_{q_2}),1\}.$$

$$f(I,J)=f(\tilde{I},\tilde{J}),$$

$$K_1\cap [A,B]={\cap }_{k=k_0}^{\infty }{G}_k\text{ and }{K}_2\cap [M,N]={\cap }_{k=k_0}^{\infty }{G}_k^{\prime }.$$

$$f(K_1\cap [A,B],K_2\cap [M,N])={\cap }_{k=k_0}^{\infty }f(G_k,G_k^{\prime }).$$

$$f(G_k,G_k^{\prime })$$

$$\frac{{\kappa }_1}{d-c}<\left|\frac{{\partial }_{y}f{|}_{(x_0,y_0)}}{{\partial }_{x}f{|}_{(x_0,y_0)}}\right|<\frac{\lambda (b-a)}{{\kappa }_2},{\partial }_{y}f{|}_{(x_0,y_0)}>0,{\partial }_{x}f{|}_{(x_0,y_0)}>0,$$

$${\partial }_{y}f(x,y)>0,{\partial }_{x}f(x,y)>0$$

$$\tilde{I}={\cup }_{j=1}^n{f}_{i_1\cdots i_{p}j}([a,b]),\tilde{J}={\cup }_{l=1}^m{g}_{j_1\cdots j_{p}l}([c,d]).$$

$$f(\tilde{I},\tilde{J})={\cup }_{j=1}^n{\cup }_{l=1}^{m}f(f_{i_1\cdots i_{p}j}([a,b]),g_{j_1\cdots j_{p}l}([c,d])).$$

$${\cup }_{l=1}^{m}f(f_{i_1\cdots i_{p}j}([a,b]),g_{j_1\cdots j_{p}l}([c,d]))$$

$$f(f_{i_1\cdots i_{p}j}(b),g_{j_1\cdots j_{p}l}(d))-f(f_{i_1\cdots i_{p}j}(a),g_{j_1\cdots j_{p}l+1}(c))\ge 0.$$

$$f(f_{i_1\cdots i_{p}j}(b),g_{j_1\cdots j_{p}l}(d))-f(f_{i_1\cdots i_{p}j}(a),g_{j_1\cdots j_{p}l+1}(c))$$

$$f(f_{i_1\cdots i_{p}j}(b),g_{j_1\cdots j_{p}l}(d))-f(f_{i_1\cdots i_{p}j}(a),g_{j_1\cdots j_{p}l+1}(c))\ge 0.$$

Full text

[TABLE]

In this paper we give an sufficient condition which guarantees that $f_{U}(K_{1},K_{2})$ contains some interiors. Our result is different from Simon and Taylor’s [13, Proposition 2.9] as we do not need the condition that the multiplication of the thickness of $K_{1}$ and $K_{2}$ is strictly greater than $1$. As a consequence, we give an application to the univoque sets in the setting of $q$-expansions.

1 Introduction

Arithmetic on some sets was pioneered by Steinhuas who proved the following classical result: for any set $E\subset\mathbb{R}^{n}$ with positive Lebesgue measure, $E-E=\{x-y:x,y\in E\}$ contains interiors. It is natural to consider the Steinhuas’ result when $E$ is of zero Lebesgue measure. Indeed, it is an important topic in fractal geometry and dynamical systems. Let $C$ be the middle-third Cantor set. Let $C*C=\{x*y:x,y\in C\}$, where $*=+,-,\cdot,\div$ (when $*=\div$, we assume $y\neq 0$). Steinhuas [14] proved that

[TABLE]

Athreya, Reznick, and Tyson [1] proved that

[TABLE]

The sum of two Cantor sets appears naturally in homoclinic bifurcations [15]. Palis [15] posed the following problem: whether it is true (at least generically) that the arithmetic sum of dynamically defined Cantor sets either has measure zero or contains an interval. This conjecture was solved in [2]. Motivated by Palis’ conjecture, it is natural to investigate when the sum of two Cantor sets contains some interiors. Newhouse [18] proved the following celebrated results. Given any two Cantor sets $C_{1}$ and $C_{2}$, if $\tau(C_{1})\tau(C_{2})>1$, where $\tau(C_{i}),i=1,2$ denotes the thickness of $C_{i},i=1,2,$ then $C_{1}+C_{2}$ contains some interiors. Motivated by Newhouse’s result, Simon and Taylor[13, Proposition 2.9] proved that for any $C^{2}$ functions $H(x,y)$ defined on some open set $U$, if two Cantor sets $C_{1}$ and $C_{2}$ have the property $\tau(C_{1})\tau(C_{2})>1$, and the partial derivatives of $H(x,y)$ are not vanishing for almost everywhere in $U$, then $H(C_{1},C_{2})=\{H(x,y):x\in C_{1},y\in C_{2}\}$ contains some intervals. Simon and Taylor’s result is elegant, however, their result is not true when $\tau(C_{1})\tau(C_{2})\leq 1$. The theme of this paper is to weaken this condition for some fractal sets.

Let $f$ be a continuous function defined on an open set $U\subset\mathbb{R}^{2}$, and $E,F$ be two non-empty sets in $\mathbb{R}$. Denote the continuous image of $f$ by

[TABLE]

Let $K_{1}$ and $K_{2}$ be the attractors of the IFS’s $\{f_{i}(x)=\lambda x+a_{i}\}_{i=1}^{n}$ and $\{g_{j}(x)=\lambda x+b_{j}\}_{j=1}^{m}$, respectively, where $a_{i},b_{j}\in\mathbb{R},0<\lambda<1.$ Let the convex hull of $K_{1}$($K_{2}$) be $[a,b]$($[c,d]$). Without loss of generality, we may assume $f_{1}(a)=a,f_{n}(b)=b$ ($g_{1}(c)=c,g_{m}(d)=d$), and

[TABLE]

Let

[TABLE]

[TABLE]

Theorem 1.1**.**

Let $K_{1},K_{2}$ be the attractors defined as above. If there is some $(x_{0},y_{0})\in(K_{1}\times K_{2})\cap U$ such that

[TABLE]

then $f_{U}(K_{1},K_{2})$ contains some interiors.

The following result can be obtained easily in terms of Theorem 1.1.

Corollary 1.2**.**

Let $K_{1},K_{2}$ be the attractors defined as above. If

[TABLE]

then for $f(x,y)=x^{\alpha}\pm y^{\alpha},\alpha\neq 0,xy,\dfrac{x}{y},$ $f_{U}(K_{1},K_{2})$ contains some interiors.

Remark 1.3**.**

We have the following remarks.

• (1)

We compare Theorem 1.1 with Simon and Taylor’s result. First, our result can be checked directedly. More importantly, we do not need the condition that the multiplication of the thickness is strictly greater than $1$.

• (2)

For the irreducible graph-directed self-similar set (inhomogeneous self-similar set), we are able to construct a homogeneous self-similar set such that it can arbitrarily approximate the irreducible graph-directed self-similar set (inhomogeneous self-similar set) in the sense of Hausdorff dimension **[9, Theorem 1.1]**, **[16, Proposition 6]**. Thus, we can consider similar problem on the irreducible graph-directed self-similar set (inhomogeneous self-similar set).

• (3)

Theorem 1.1 is very useful to many other fractal sets, for instance, the set of points with multiple codings, some attractors generated by the open dynamics and so forth. The usage of Theorem 1.1 is as follows: for a given fractal set, we firstly find a sub self-similar set of this fractal set (if it is an inhomogeneous self-similar set, then we use the second remark above), then we utilize Theorem 1.1 to obtain partial results. We emphasize here that **[9, Theorem 1.1]**, **[16, Proposition 6]** is very helpful to analyze the general fractal sets.

We give an application to $q$-expansions. Let $1<q<2$. It is well-known that for every $x\in\left[0,\dfrac{1}{q-1}\right]$ there is some $(a_{n})\in\{0,1\}^{\mathbb{N}}$ such that

[TABLE]

We call $(a_{n})$ an $q$-expansion of $x$. Typically, $x$ has multiple expansions [17, 5]. If $x$ has a unique expansion, then we call $x$ a univoque point. Write $U_{q}$ for the set of points with unique expansions. Sidorov proved in [18] that for any $q\geq T_{3}$, then

[TABLE]

where $T_{3}$ is the Pisot number satisfying $x^{3}=x^{2}+x+1.$ He also posed a question that finding the smallest base $q$ such that $U_{q}+U_{q}=\left[0,\dfrac{2}{q-1}\right]$. In [6], Dajani, Komornik, Kong and Li proved that $\mathcal{U}(x)*\mathcal{U}(x)$ contains interiors, where $*=+,-,\cdot,\div$, and $\mathcal{U}(x)$ is the set of bases for which the expansion of $x\in(0,1]$ is unique. The main tool of proving the above two results is the Newhouse thickness theorem. Motivated by their results, it is natural to consider the arithmetic on the univoque set $U_{q}$. To the best of our knowledge, there are very few results concerning with $f_{U}(U_{q},U_{q})$, where $f(x,y)$ is a general function. The following results are corollaries of Theorem 1.1.

Corollary 1.4**.**

Let $q_{*}\approx 1.8019$ be the appropriate root of

[TABLE]

For any $q\in(q_{*},2)$, if there is some $(x_{0},y_{0})\in(K_{q}\times K_{q})\cap U$ such that

[TABLE]

where $\lambda=q^{-2}$ and $K_{q}$ is the attractor of the IFS

[TABLE]

then $f_{U}(U_{q},U_{q})$ contains an interior.

Corollary 1.5**.**

If $q\in(q_{*},2)$, then

[TABLE]

have interiors.

It is natural to consider the dimension of $f_{U}(U_{q},U_{q})$. The following results are indeed some corollaries of Bárány [3], Peres and Shmerkin[16].

Proposition 1.6**.**

For any $1<q<2$,

[TABLE]

For two different bases $q_{1},q_{2}\in(1,2)$ with the property

[TABLE]

then

[TABLE]

The paper is arranged as follows. In section 2, we give the proofs of main results. In section 3, we give some remarks.

2 Proof of Main results

In this section, we shall give the proofs of the main theorems of this paper.

2.1 Proof of Theorem 1.1

Before, we prove Theorem 1.1, we recall some definitions. For simplicity, we only introduce the definitions for $K_{1}$. For any $(i_{1},\cdots,i_{k})\in\{1,2,\cdots,n\}^{k}$, we call $f_{i_{1},\cdots,i_{k}}([a,b])=(f_{i_{1}}\circ\cdots\circ f_{i_{k}})([a,b])$ a basic interval of rank $k,$ which has length $\lambda^{k}(b-a)$. In what follows, we use the following notation $f_{i_{1}}\circ\cdots\circ f_{i_{k}}=f_{i_{1},\cdots,i_{k}}.$ Denote by $H_{k}$ the collection of all these basic intervals of rank $k$ with respect to $\{f_{i}\}_{i=1}^{n}$. Let $J\in H_{k}$, define $\widetilde{J}=\cup_{i=1}^{n}I_{k+1,i}$ , where $I_{k+1,i}\in H_{k+1}$ and $I_{k+1,i}\subset J$ for $i=1,2,\cdots n$. Let $[A,B]\subset[a,b]$, where $A$ and $B$ are the left and right endpoints of some basic intervals in $H_{k}$ for some $k\geq 1$, respectively. $A$ and $B$ may not be in the same basic interval. Let $F_{k}$ be the collection of all the basic intervals in $[A,B]$ with length $\lambda^{k}(b-a),k\geq k_{0}$ for some $k_{0}\in\mathbb{N}^{+}$, i.e. the union of all the elements of $F_{k}$ is denoted by $G_{k}=\cup_{i=1}^{t_{k}}I_{k,i}$, where $t_{k}\in\mathbb{N}^{+}$, $I_{k,i}\in H_{k}$ and $I_{k,i}\subset[A,B]$. Clearly, by the definition of $G_{k}$, it follows that $G_{k+1}\subset G_{k}$ for any $k\geq k_{0}.$ Similarly, we can define the basic intervals for $K_{2}$. Let $M$ and $N$ are the left and right endpoints of some basic intervals in $H_{k}^{{}^{\prime}}$ with respect to $K_{2}$. Denote by $G_{k}^{\prime}$ the union of all the basic intervals with length $\lambda^{k}(d-c)$ in the interval $[M,N]$, i.e. $G_{k}^{\prime}=\cup_{j=1}^{t_{k}^{\prime}}J_{k,j}^{\prime}$, where $t_{k}^{\prime}\in\mathbb{N}^{+}$, $J_{k,j}^{\prime}\in H_{k}^{\prime}$ and $J_{k,j}^{\prime}\subset[M,N]$. The following lemma is crucial to our analysis.

Lemma 2.1**.**

Let $f:U\rightarrow\mathbb{R}$ be a continuous function. Suppose $A$ and $B$ ($M$ and $N$) are the left and right endpoints of some basic intervals in $H_{k_{0}}$ ($H_{k_{0}}^{\prime}$) for some $k_{0}\geq 1$ respectively such that $[A,B]\times[M,N]\subset U.$ Then $K_{1}\cap[A,B]=\cap_{k={k_{0}}}^{\infty}G_{k}$, and $K_{2}\cap[M,N]=\cap_{k={k_{0}}}^{\infty}G_{k}^{\prime}$. Moreover, if for any $k\geq k_{0}$ and any two basic intervals $I\subset G_{k}$, $J\subset G_{k}^{\prime}$ such that

[TABLE]

then $f(K_{1}\cap[A,B],K_{2}\cap[M,N])=f(G_{k_{0}},G_{k_{0}}^{\prime}).$

Proof.

By the construction of $G_{k}$ ($G_{k}^{\prime}$), i.e. $G_{k+1}\subset G_{k}$ ($G_{k+1}^{\prime}\subset G_{k}^{\prime}$) for any $k\geq k_{0}$, it follows that

[TABLE]

The continuity of $f$ yields that

[TABLE]

In terms of the relation $G_{k+1}=\widetilde{G_{k}}$, $G_{k+1}^{\prime}=\widetilde{G_{k}^{\prime}}$ and the condition in the lemma, it follows that

[TABLE]

Therefore, $f(K_{1}\cap[A,B],K_{2}\cap[M,N])=f(G_{k_{0}},G_{k_{0}}^{\prime}).$ ∎

Theorem 2.2**.**

Suppose that $\partial_{x}f,\partial_{y}f$ are continuous. If there is some $(x_{0},y_{0})\in(K_{1}\times K_{2})\cap U$ such that

[TABLE]

then $f_{U}(K_{1},K_{2})$ contains some interiors.

Proof.

Since $\partial_{x}f,\partial_{y}f$ are continuous, and $\partial_{y}f|_{(x_{0},y_{0})}>0,\partial_{x}f|_{(x_{0},y_{0})}>0$, it follows that there is some neighbourhood of $(x_{0},y_{0})$, denoted by $V$, such that

[TABLE]

for any $(x,y)\in V.$ Let $I=f_{i_{1}\cdots i_{p}}([a,b])$ and $J=g_{j_{1}\cdots j_{p}}([c,d])$ be two basic intervals of $H_{k}$ and $H_{k}^{\prime}$, respectively. We assume that $I\times J\subset V$. By the definition $\widetilde{I}$ and $\widetilde{J}$, we have

[TABLE]

Therefore,

[TABLE]

For any $1\leq j\leq n$, we shall prove that

[TABLE]

is an interval. In fact, by the conditions $\partial_{y}f|_{(x_{0},y_{0})}>0,\partial_{x}f|_{(x_{0},y_{0})}>0$, it remains to prove that

[TABLE]

Note that

[TABLE]

for any $(x,y)\in I\times J\subset V$. By the condition $\dfrac{\partial_{y}f|_{(x_{0},y_{0})}}{\partial_{x}f|_{(x_{0},y_{0})}}<\dfrac{\lambda(b-a)}{\kappa_{2}}$ and the continuity of $\partial_{x}f,\partial_{y}f$, it follows that

[TABLE]

Next, we want to show that

[TABLE]

is an interval for any $1\leq j\leq n-1$. Indeed, it suffices to prove that

[TABLE]

Clearly if $f_{j}(b)-f_{j+1}(a)\geq 0$, then

[TABLE]

If $f_{j}(b)-f_{j+1}(a)<0$, i.e. $j\in D_{1}$, then by the condition $\dfrac{\kappa_{1}}{b-a}<\dfrac{\partial_{y}f|_{(x_{0},y_{0})}}{\partial_{x}f|_{(x_{0},y_{0})}}$ and the continuity of $\partial_{x}f,\partial_{y}f$, we also have

[TABLE]

Therefore, we have proved that $f(I,J)=f(\widetilde{I},\widetilde{J})$. In terms of Lemma 2.1, it follows that $f_{U}(K_{1},K_{2})$ contains some interiors. ∎

Similarly, we can prove the following result. We left it to the readers.

Theorem 2.3**.**

Suppose that $\partial_{x}f,\partial_{y}f$ are continuous. If there is some $(x_{0},y_{0})\in(K_{1}\times K_{2})\cap U$ such that

[TABLE]

then $f_{U}(K_{1},K_{2})$ contains some interiors.

Proof of Theorem 1.1.

By Theorems 2.2 and 2.3, it remains to prove the following two cases.

• (1)

If $\partial_{y}f|_{(x_{0},y_{0})}<0,\partial_{x}f|_{(x_{0},y_{0})}<0,$ then we let $F_{1}(x,y)=-f(x,y)$ and use Theorem 2.2.

• (2)

If $\partial_{y}f|_{(x_{0},y_{0})}>0,\partial_{x}f|_{(x_{0},y_{0})}<0,$ then we let $F_{2}(x,y)=-f(x,y)$, and make use of Theorem 2.3.

∎

2.2 Proof of Theorem 1.4

Now, we prove Theorem 1.4. Our idea is simple, i.e. for any $q\in(q_{*},2)$, we shall construct some self-similar set, denoted by $K_{q}$, contained in $U_{q}$ such that $f_{U}(K_{q},K_{q})$ contains some interiors. Therefore, $f_{U}(U_{q},U_{q})$ has some interiors. Before we construct the set $K_{q}$ we give some classical results of unique expansions. The following theorem characterizes the criteria of the unique expansions, the proof of this result can be found in [8] or some references therein.

Theorem 2.4**.**

Let $(a_{n})_{n=1}^{\infty}$ be an expansion of $x\in[0,(q-1)^{-1}]$. Then $(a_{n})_{n=1}^{\infty}\in\widetilde{U}_{q}$ if and only if

[TABLE]

wherever $a_{k}=0$,

[TABLE]

wherever $a_{k}=1$, where $(\eta_{n})_{n=1}^{\infty}$ is the quasi-greedy expansions of $1$, $\widetilde{U}_{q}$ denotes all the unique expansions in base $q$, and $``<"$ means the lexicographic order.

Lemma 2.5**.**

Let $q_{*}\approx 1.8019$ be the appropriate root of

[TABLE]

Then for any $q\in(q_{*},2)$,

[TABLE]

where $K_{q}$ is the attractor of the following IFS

[TABLE]

Proof.

In base $q_{*}$, the quasi-greedy expansion of $1$ is $11(01)^{\infty}.$ Therefore, by Theorem 2.4, we have that for any $q\in(q_{*},2)$,

[TABLE]

Thus, we can construct a self-similar set $K_{q}$ by the coding space $\{(01),(10)\}^{\infty}$, namely,

[TABLE]

where $K_{q}$ is the attractor of the IFS

[TABLE]

∎

Now, we are able to prove Corollary 1.4.

Proof of Corollary 1.4.

For the self-similar set $K_{q}$, note that $a=\dfrac{1}{q^{2}-1},b=\dfrac{q}{q^{2}-1}$, $r=q^{-2}$. Therefore, the condition

[TABLE]

is exactly the following condition

[TABLE]

where $\lambda=q^{-2}$. Therefore, we prove Corollary 1.4. ∎

Proof of Corollary 1.5.

For $f(x,y)=xy\mbox{ or }x^{2}+y^{2}\mbox{ or }x^{2}-y^{2},y/x$

[TABLE]

Without loss of generality, we only consider the following case as for the remaining case, the discussion is analogous. Suppose

[TABLE]

We take $(x_{0},y_{0})=\left(\dfrac{1}{q^{2}-1},\dfrac{q}{q^{2}-1}\right)\in K_{q}\times K_{q}$. It is easy to check that in this case

[TABLE]

for any $q\in(q_{*},2)$, where $\lambda=q^{-2}.$ Therefore, Corollary 1.5 follows Corollary 1.4. ∎

2.3 Proof of Proposition 1.6

Bárány [3] proved the following result.

Theorem 2.6**.**

Let $\Lambda$ be an arbitrary self-similar set in $\mathbb{R}^{2}$ not contain in any line. Suppose that $g:\mathbb{R}^{2}\to\mathbb{R}$ is a $C^{2}$ map such that

[TABLE]

for any $(x,y)\in\Lambda$. Then

[TABLE]

In terms of this result, we can prove Proposition 1.6.

Proof of Proposition 1.6.

First, we prove that for any $1<q<2$,

[TABLE]

It remains to prove that the formula is correct for any $q_{KL}<q<2$, where $q_{KL}$ is the Komornik-Loreti constant. As for any $q\in(1,q_{KL}]$, $\dim_{H}(U_{q})=0$ (see [10]), and the fact that for any $C^{2}$ map $g$, we always have

[TABLE]

In our proposition, $g(x,y)=xy,x^{2}\pm y^{2},x/y$. In what follows, we always assume $g(x,y)$ is one of the four functions. For $f(x,y)=x/y$, we may adjust the proof if necessary. Let $\widetilde{U_{q}}$ be the associated coding space of $U_{q}$. For any $\epsilon>0$, we may find a subshift of finite type, denoted by $\widetilde{U_{q_{0}}},$ with transitivity condition [4] such that $\widetilde{U_{q_{0}}}\subset\widetilde{U_{q}}$ and that

[TABLE]

where $\pi_{q}(\cdot)$ means the natural projection of the coding space in base $q$. Note that $\pi_{q}(\widetilde{U_{q_{0}}})$ is a graph-directed self-similar set. We denote it by $(K_{1},K_{2},\cdots,K_{n})$. By the transitivity condition, it follows that

[TABLE]

for any $1\leq i\leq n.$ Given $K_{i}$, by Theorem [9, Theorem 1.1], for any $\epsilon>0$, there is some self-similar set $K$ such that

[TABLE]

Therefore, for any $q_{KL}<q<2$, and any $\epsilon>0$ there exists some homogeneous self-similar set $K\subset U_{q}$ such that

[TABLE]

Note that $K\times K$ is a self-similar set in $\mathbb{R}^{2}$ which is not contained in a line. By Theorem 2.6

[TABLE]

Here if $g(x,y)=x/y$, we may replace $U_{q}$ by $\left[\dfrac{2-q}{q-1},1\right]\cap U_{q}$. As for this case, $0\in U_{q}$ and we need to avoid this point if we use Theorem 2.6. The rest proof is the same. Therefore,

[TABLE]

Letting $\epsilon\to 0$, and we obtain that

[TABLE]

and subsequently we have proved that

[TABLE]

The proof of the remaining formula is similar. With a similar discussion, for any $q_{1},q_{2}\in(q_{KL},2)$ we have

[TABLE]

where $P$ denotes the projection to the $y$-axis through the angle $3\pi/4.$ Thus

[TABLE]

In order to show that $\dim_{H}((U_{q_{1}}+U_{q_{2}}))\geq\min\{\dim_{H}(U_{q_{1}})+\dim_{H}(U_{q_{2}}),1\}$. Arbitrarily fix an $\epsilon>0$. As in the proof of

[TABLE]

one can take graph-directed self-similar sets $\pi_{q_{1}}(\widetilde{U_{1}})$ and $\pi_{q_{2}}(\widetilde{U_{2}})$ such that $\pi_{q_{1}}(\widetilde{U_{1}})\subseteq U_{q_{1}}$, $\pi_{q_{2}}(\widetilde{U_{2}})\subseteq U_{q_{2}}$ and

[TABLE]

By [9, Remark 1.2], there exist some homogeneous self-similar sets $K_{q_{1}}$ and $K_{q_{2}}$ with similarity ratios $1/q_{1}$ and $1/q_{2}$, respectively, such that

[TABLE]

and that

[TABLE]

By [16, Theorem 2], we have

[TABLE]

Therefore,

[TABLE]

Therefore,

[TABLE]

∎

3 Final remarks

It would be interesting if one can give the exact form of $f_{U}(K_{1},K_{2})$ for some functions. For instance, what is the exact structure of $C\cdot C$? We shall give an answer to this question in another paper.

Acknowledgements

The work is supported by National Natural Science Foundation of China (Nos.11701302,

11671147). The work is also supported by K.C. Wong Magna Fund in Ningbo University. Kan Jiang would like to thank Karma Dajani, Derong Kong and Wenxia Li for some suggestions on the previous versions of the manuscript.