# Multiple expansions of real numbers with digits set \(\left\{ 0,1,q\right\} \)

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## Abstract

*q*with digits set \(\left\{ 0,1,q\right\} \). Let \({{\mathcal {U}}}_q\) be the set of points which have a unique

*q*-expansion. For \(k=2, 3,\ldots ,\aleph _0\) let \(\mathcal {B}_k\) be the set of bases \(q>1\) for which there exists

*x*having precisely

*k*different

*q*-expansions, and for \(q\in \mathcal {B}_k\) let \({{\mathcal {U}}}_q^{(k)}\) be the set of all such

*x*’s which have exactly

*k*different

*q*-expansions. In this paper we show that

*q*-expansions has full Hausdorff dimension.

## Keywords

Unique expansion Multiple expansion Countable expansion Hausdorff dimension## Mathematics Subject Classification

Primary 11A63 Secondary 10K50 11K55 37B10## 1 Introduction

Expansions in non-integer bases were pioneered by Rényi [18] and Parry [16]. Unlike integer base expansions, for a given \(\beta \in (1, 2)\), it is well-known that typically a real number \(x\in I_\beta :=[0, 1/(\beta -1)]\) has a continuum of \(\beta \)-expansions with digits set \(\left\{ 0, 1\right\} \) (cf. [2, 19]), i.e., for Lebesuge almost every \(x\in I_\beta \) there exist a continuum of zero-one sequences \((x_i)\) such that \(x=\sum _{i=1}^\infty x_i/\beta ^i\). However, there still exist \(x\in I_\beta \) having a unique \(\beta \)-expansion (cf. [5, 10, 13]). Denote by \({{\mathcal {U}}}_\beta \) the set of all \(x\in I_\beta \) with a unique \(\beta \)-expansion. De Vries and Komornik [3] investigated the topological properties of \({{\mathcal {U}}}_\beta \). Komornik et al. [12] considered the Hausdorff dimension of \({{\mathcal {U}}}_\beta \), and concluded that the dimension function \(\beta \mapsto \dim _H{{\mathcal {U}}}_\beta \) behaves like a Devil’s staircase. Interestingly, for any \(k=2,3,\ldots \) or \(\aleph _0\) Erdős et al. [6, 7] showed that there exist \(\beta \in (1,2)\) and \(x\in I_\beta \) such that *x* has precisely *k* different \(\beta \)-expansions. For more information on expansions in non-integer bases we refer to [1, 21, 23], and the surveys [4, 11, 20].

*q*

*-expansion*of

*x*, if

*digits set*\(\left\{ 0,1,q\right\} \) also depends on the base

*q*.

*q*-expansion. Then \(E_q\) is the attractor of the

*iterated function system*(IFS)

*self-similar set with overlaps*. Ngai and Wang [15] gave the Hausdorff dimension of \(E_q\):

*q*-expansion. In this paper, we investigate the set of points in \(E_q\) having multiple

*q*-expansions.

*q*-expansion, and denote by \( {{\mathcal {U}}}'_q\) the set of all

*q*-expansions corresponding to elements of \({{\mathcal {U}}}_q\).

In this paper we will describe the sizes of the sets \(\mathcal {B}_k\) and \({{\mathcal {U}}}_q^{(k)}\). Our first result is on the set \(\mathcal {B}_k\) for \(k= 1,2,\ldots ,\aleph _0\) or \(2^{\aleph _0}\). Clearly, when \(k=1\) we have \(\mathcal {B}_1=(1,\infty )\), since 0 always has a unique *q*-expansion for any \(q>1\). When \(k= 2,3,\ldots ,\aleph _0\) or \(2^{\aleph _0}\) we have the following

### Theorem 1

By Theorem 1 it follows that for \(q\in [2,q_c]\), any \(x\in E_q\) can only have a unique *q*-expansion, countably infinitely many *q*-expansions, or a continuum of *q*-expansions.

When \(k=1\), the following theorem for the *univoque set*\({{\mathcal {U}}}_q={{\mathcal {U}}}_q^{(1)}\) was proven in [24].

### Theorem 1.1

- (i)
If \(q\in (1, q_c]\), then \({{\mathcal {U}}}_q=\left\{ 0,q/(q-1)\right\} \).

- (ii)
If \(q\in (q_c, q^*)\), then \({{\mathcal {U}}}_q\) contains a continuum of points.

- (iii)
If \(q\in [q^*,\infty )\), then \(\dim _H{{\mathcal {U}}}_q=\log q_c/\log q\).

Our second result complements Theorem 1.1, and shows that there is no difference between the Hausdorff dimensions of \({{\mathcal {U}}}_q^{(k)}\) and \({{\mathcal {U}}}_q\).

### Theorem 2

- (i)
\(\dim _H{{\mathcal {U}}}_{q}>0\) if and only if \(q>q_c\).

- (ii)
For any integer \(k\ge 2\) and any \(q\in \mathcal {B}_{k}\) we have \( \dim _H{{\mathcal {U}}}_q^{(k)}=\dim _H{{\mathcal {U}}}_q. \)

As a result of Theorem 2 it follows that \(q_c\) is indeed the *critical base*, in the sense that \({{\mathcal {U}}}_q^{(k)}\) has positive Hausdorff dimension if \(q>q_c\), while \({{\mathcal {U}}}_q^{(k)}\) has zero Hausdorff dimension if \(q\le q_c\). In fact, by Theorems 1 and 1.1 (i) it follows that for \(q\le q_c\) the set \({{\mathcal {U}}}_q=\left\{ 0, q/(q-1)\right\} \) and \({{\mathcal {U}}}_q^{(k)}=\emptyset \) for any integer \(k\ge 2\).

Our final result focuses on the sizes of \({{\mathcal {U}}}_q^{(\aleph _0)}\) and \({{\mathcal {U}}}_q^{(2^{\aleph _0})}\).

### Theorem 3

- (i)
Let \(q\in \mathcal {B}_{\aleph _0}{\setminus }(q_c, q^*)\). Then \({{\mathcal {U}}}_q^{(\aleph _0)}\) is countably infinite.

- (ii)
For any \(q>1\) we have \( \dim _H{{\mathcal {U}}}_q^{(2^{\aleph _0})}=\dim _H E_q. \)

### Remark 1.2

The rest of the paper is arranged as follows. In Sect. 2 we recall some properties of unique *q*-expansions. The proof of Theorem 1 for the sets \(\mathcal {B}_k\) will be presented in Sect. 3, and the proofs of Theorems 2 and 3 for the sets \({{\mathcal {U}}}_q^{(k)}\) will be given in Sects. 4 and 5, respectively. Finally, in Sect. 6 we give some examples and end the paper with some questions.

## 2 Unique expansions

Given \(q>1\), let \(\left\{ 0,1,q\right\} ^{\mathbb {N}}\) be the set of all infinite sequences \((d_i)\) over the alphabet \(\left\{ 0,1,q\right\} \). By a word \({\mathbf {c}}\) we mean a finite string of digits \({\mathbf {c}}=c_1\ldots c_n\) with each digit \(c_i\in \left\{ 0,1,q\right\} \). For two words \({\mathbf {c}}=c_1\ldots c_m\) and \(\mathbf d=d_1\ldots d_n\), we denote by \({\mathbf {c}}\mathbf d=c_1\ldots c_md_1\ldots d_n\) their concatenation. For a positive integer *k* we write \({\mathbf {c}}^k={\mathbf {c}}\cdots {\mathbf {c}}\) for the *k*-fold concatenation of \({\mathbf {c}}\) with itself. Furthermore, we write \({\mathbf {c}}^\infty ={\mathbf {c}}{\mathbf {c}}\cdots \) the infinite periodic sequence with periodic block \({\mathbf {c}}\). Throughout the paper we will use lexicographical ordering \(\prec , \preccurlyeq , \succ \) and \(\succcurlyeq \) between sequences. More precisely, for two sequences \((c_i), (d_i)\in \left\{ 0,1,q\right\} ^{{\mathbb {N}}}\) we say \((c_i)\prec (d_i)\) or \((d_i)\succ (c_i)\) if there exists an integer \(n\ge 1\) such that \(c_1\ldots c_{n-1}=d_1\ldots d_{n-1}\) and \(c_n<d_n\). Furthermore, we say \((c_i)\preccurlyeq (d_i)\) if \((c_i)\prec (d_i)\) or \((c_i)=(d_i)\).

*q*-expansion, and \({{\mathcal {U}}}_q'\) is the set of corresponding

*q*-expansions. Then

### Lemma 2.1

*quasi-greedy*

*q*-expansion of \(q-1\), i.e., the lexicographically largest

*q*-expansion of \(q-1\) with infinitely many non-zero digits. We emphasize that \(\alpha (q)\) is well-defined for \(q\in (1, q^*]\). By (2.1) and a direct calculation one can verify that

### Lemma 2.2

### Lemma 2.3

The map \(q\rightarrow \Phi (\alpha (q))\) is strictly increasing in \((1,q^*]\).

By (2.2) and Lemma 2.3 it follows that for any \(q\in (q_c, q^*)\) we have \(q1^\infty \prec \alpha (q)\prec q^\infty \).

## 3 Proof of Theorem 1

In this section we will investigate the set \(\mathcal {B}_k\) of bases \(q>1\) in which there exists \(x\in E_q\) having *k* different *q*-expansions. Excluding the trivial case for \(k=1\) that \(\mathcal {B}_1=(1,\infty )\) we consider \(\mathcal {B}_k\) for \(k= 2,3,\ldots ,\aleph _0\) or \(2^{\aleph _0}\).

The following lemma was established in [24, Theorem 4.1] and [9, Theorem 1.1].

### Lemma 3.1

- (i)
If \(q\in (1,2)\), then any \(x\in E_q\) has either a unique

*q*-expansion, or a continuum of*q*-expansions. - (ii)
If \(q=2\), then any \(x\in E_q\) can only have a unique

*q*-expansion, countably infinitely many*q*-expansions, or a continuum of*q*-expansions.

*switch region*, since any \(x\in S_q\) has at least two

*q*-expansions. More precisely, any \(x\in \phi _0(E_q)\cap \phi _1(E_q)\) has at least two

*q*-expansions: one begins with the digit 0 and one begins with the digit 1. Accordingly, any \(x\in \phi _1(E_q)\cap \phi _q(E_q)\) also has at least two

*q*-expansions: one starts with the digit 1 and one starts with the digit

*q*. We point out that the union in (3.1) is disjoint if \(q>2\). In particular, for \(q>q^*\) the intersection \(\phi _1(E_q)\cap \phi _q(E_q)=\emptyset \).

*q*-expansions of

*x*, i.e.,

*q*

*-null infinite point*if

*x*has an expansion \((d_i)\in \left\{ 0,1,q\right\} ^\mathbb {N}\) such that whenever

*q*-null infinite point has countably infinitely many

*q*-expansions.

First we consider the set \(\mathcal {B}_{\aleph _0}\), which is based on the following characterization (cf. [1, 23]).

### Lemma 3.2

\(q\in \mathcal {B}_{\aleph _0}\) if and only if \(S_q\) contains a *q*-null infinite point.

### Lemma 3.3

\(\mathcal {B}_{\aleph _0}=[2,\infty )\).

### Proof

By Lemma 3.1 we have \(\mathcal {B}_{\aleph _0}\subseteq [2,\infty )\) and \(2\in \mathcal {B}_{\aleph _0}\). So, it suffices to prove \((2,\infty )\subseteq \mathcal {B}_{\aleph _0}\).

*q*-null infinite point. Note that \((10^\infty )_q=(0q0^\infty )_q\). Then by the words substitution \(10\sim 0q\) it follows that all expansions \(1^k0 q^\infty , k\ge 0,\) are

*q*-expansions of

*x*, i.e.,

*x*is a

*q*-null infinite point. Hence, by Lemma 3.2 we have \(q\in \mathcal {B}_{\aleph _0}\), and therefore \((2,\infty )\subseteq \mathcal {B}_{\aleph _0}\). \(\square \)

Now we turn to describe the set \(\mathcal {B}_k\). By Lemma 3.1 it follows that \(\mathcal {B}_k\subseteq (2,\infty )\) for any \(k\ge 2\). First we consider \(\mathcal {B}_2\) and need the following

### Lemma 3.4

### Proof

*q*-expansions, say

*x*has exactly two different

*q*-expansions. So, \(q\in \mathcal {B}_2\). \(\square \)

Recall from (2.2) that \(q_c\approx 2.32472\) and \(q^*=(3+\sqrt{5})/2\) admit the quasi-greedy expansions \(\alpha (q_c)=q_c1^\infty \) and \(\alpha (q^*)=(q^*)^\infty .\) In the following lemma we describe the set \(\mathcal {B}_2\).

### Lemma 3.5

\(\mathcal {B}_2=(q_c,\infty )\).

### Proof

*q*must satisfy one of the following equations

*q*-expansions. This implies that \((q^*,\infty )\subseteq \mathcal {B}_2\).

*q*-expansions. So, \((q_c,q^*]\subseteq \mathcal {B}_2\), and the proof is complete. \(\square \)

### Lemma 3.6

\(\mathcal {B}_k=(q_c,\infty )\) for any \(k\ge 3\).

### Proof

*k*different

*q*-expansions. Since \(q>2\), the union in (3.1) is disjoint. This implies that there exists a word \(d_1\ldots d_n\) such that

*q*-expansions. So, \(q\in \mathcal {B}_2\). Hence, \(\mathcal {B}_k\subseteq \mathcal {B}_2\) for any \(k\ge 3\).

*k*different

*q*-expansions. We will prove this by induction on

*k*.

*k*different

*q*-expansions. Now we consider \(x_{k+1}\), which can be written as

*k*different

*q*-expansions. Then \(x_{k+1}\) has at least \(k+1\) different

*q*-expansions. On the other hand, since \(q>q^*>2\), the union in (3.1) is disjoint. Then

*q*-expansions. By induction this proves the claim, and hence \((q^*, \infty )\subseteq \mathcal {B}_k\) for all \(k\ge 3\).

*k*different

*q*-expansions. Again, this will be proven by induction on

*k*.

*q*-expansion. Suppose \(y_k\) has exactly

*k*different

*q*-expansions. Now we consider

*k*different

*q*-expansions. This implies that \(y_{k+1}\) has at least \(k+1\) different

*q*-expansions. On the other hand, note that \(q>q_c>2\), and therefore the union in (3.1) is disjoint. So, \( y_{k+1}\in \phi _0(E_q)\cap \phi _1(E_q){\setminus }\phi _q(E_q), \) which implies that \(y_{k+1}\) indeed has \(k+1\) different

*q*-expansions. By induction this proves the claim, and then \((q_c, q^*]\subseteq \mathcal {B}_k\) for all \(k\ge 3\). This completes the proof. \(\square \)

### Proof of Theorem 1

*x*indeed has a continuum of different

*q*-expansions. \(\square \)

## 4 Proof of Theorem 2

For \(q>1\) and \(k\in \mathbb {N}\) we recall that \({{\mathcal {U}}}_q^{(k)}\) is the set of \(x\in [0, q/(q-1)]\) having precisely *k* different *q*-expansions. In this section we are going to investigate the Hausdorff dimension of \({{\mathcal {U}}}_q^{(k)}\). First we show that \(q_c\approx 2.32472\) is the critical base for \({{\mathcal {U}}}_q\).

### Lemma 4.1

Let \(q>1\). Then \(\dim _H{{\mathcal {U}}}_q>0\) if and only if \(q>q_c\).

### Proof

In the following we will consider the Hausdorff dimension of \({{\mathcal {U}}}_q^{(k)}\) for any \(k\ge 2\), and prove \(\dim _H{{\mathcal {U}}}_q^{(k)}=\dim _H{{\mathcal {U}}}_q\). The upper bound of \(\dim _H{{\mathcal {U}}}_q^{(k)}\) is easy.

### Lemma 4.2

Let \(q>1\). Then \(\dim _H{{\mathcal {U}}}_q^{(k)}\le \dim _H{{\mathcal {U}}}_q\) for any \(k\ge 2\).

### Proof

*follower set*in \({{\mathcal {U}}}_q'\) generated by the word 1, and let \(F_q(1)\) be the set of \(x\in E_q\) which have a

*q*-expansion in \(F_q'(1)\), i.e., \(F_q(1)=\{((d_i))_q: (d_i)\in F_q'(1)\}.\)

### Lemma 4.3

Let \(q>q_c\). Then \(\dim _H{{\mathcal {U}}}_q^{(k)}\ge \dim _H F_q(1)\) for any \(k\ge 1\).

### Proof

*k*that \(x_k\) has exactly

*k*different

*q*-expansions.

*k*different

*q*-expansions. Now we consider \(x_{k+1}\), which can be expanded as

*k*different

*q*-expansions. This implies that \(x_{k+1}\) has at least \(k+1\) different

*q*-expansions. On the other hand, since \(q>q_c>2\), it gives that the union in (3.1) is disjoint. So, \(x_{k+1}\in \phi _0(E_q)\cap \phi _1(E_q){\setminus }\phi _q(E_q),\) which implies that \(x_{k+1}\) indeed has \(k+1\) different

*q*-expansions.

By induction this proves \(x_k\in {{\mathcal {U}}}_q^{(k)}\) for all \(k\ge 1\). Since \(x_k\) was taken arbitrarily from \(\Lambda _q^k\), we conclude that \(\Lambda _q^k\subseteq {{\mathcal {U}}}_q^{(k)}\) for any \(k\ge 1\). The proof is complete. \(\square \)

### Lemma 4.4

Let \(q>q_c\). Then \(\dim _H F_q(1)\ge \dim _H{{\mathcal {U}}}_q\).

### Proof

## 5 Proof of Theorem 3

In this section we will consider the set \({{\mathcal {U}}}_q^{(\aleph _0)}\) which consists of all \(x\in E_q\) having countably infinitely many *q*-expansions.

### Lemma 5.1

For any \(q\in \mathcal {B}_{\aleph _0}\) the set \({{\mathcal {U}}}_q^{(\aleph _0)}\) contains infinitely many points.

### Proof

*q*-null infinite points, and thus \(z_k\in {{\mathcal {U}}}_q^{(\aleph _0)}\).

If \(q>2\), then by the proof of Lemma 3.3 it yields that \(z_1=(0q^\infty )_q\) is a *q*-null infinite point. Moreover, note that \( z_k=\phi _0^{k-1}(z_1)\notin S_q \) for any \(k\ge 2\). This implies that all of these points \(z_k, k\ge 1\), are *q*-null infinite points. So, \( \left\{ z_k: k\ge 1\right\} \subseteq {{\mathcal {U}}}_q^{(\aleph _0)}. \)

*q*-null infinite point. In fact, all of the

*q*-expansions of \(z_k=(0^kq^\infty )_q\) are of the form

By Lemma 5.1 it follows that \({{\mathcal {U}}}_q^{(\aleph _0)}\) is at least countably infinite for any \(q\in \mathcal {B}_{\aleph _0}=[2,\infty )\). In the following lemma we show that \({{\mathcal {U}}}_q^{(\aleph _0)}\) is indeed countably infinite if \(q\ge q^*\).

### Lemma 5.2

Let \(q\ge q^*\). Then \({{\mathcal {U}}}_q^{(\aleph _0)}\) is at most countable.

### Proof

*x*has a

*q*-expansion \((d_i)\) such that

Note by the proof of Lemma 4.4 that \({{\mathcal {U}}}_q'\subseteq X_A'\), where \(X_A'\) is a sub-shift of finite type over the state \(\left\{ 0,1,q\right\} \) with adjacency matrix *A* defined in (4.2). Moreover, \(X_A'{\setminus } {{\mathcal {U}}}_q'\) is at most countable (cf. [24, Theorem 3.4]). Note that the expansion \((d_i)\) of \(x\in {{\mathcal {U}}}_q^{(\aleph _0)}\) does not end in \({{\mathcal {U}}}_q'\). Then it suffices to prove that the sequence \((d_i)\) must end in \(X_A'\).

Suppose on the contrary that \((d_i)\) does not end in \(X_A'\). Then by (4.2) the word 0*q* or 10 occurs infinitely many times in \((d_i)\). Using the word substitution \(0q\sim 10\) this implies that \(x=((d_i))_q\) has a continuum of *q*-expansions, leading to a contradiction with \(x\in {{\mathcal {U}}}_q^{(\aleph _0)}\). \(\square \)

Furthermore, we can prove that \({{\mathcal {U}}}_q^{(\aleph _0)}\) is also countably infinite for \(q\in [2,q_c]\).

### Lemma 5.3

Let \(q\in [2,q_c]\). Then \({{\mathcal {U}}}_q^{(\aleph _0)}\) is at most countable.

### Proof

When \(q\in (q_c, q^*)\), one might expect that \({{\mathcal {U}}}_q^{(\aleph _0)}\) is also countably infinite. Unfortunately, we are not able to prove this. Instead, we show that the Hausdorff dimension of \({{\mathcal {U}}}_q^{(\aleph _0)}\) is strictly smaller than \(\dim _H E_q=1\).

### Lemma 5.4

For \(q\in (q_c, q^*)\) we have \( \dim _H{{\mathcal {U}}}_q^{(\aleph _0)}\le \dim _H{{\mathcal {U}}}_q<1\).

### Proof

*A*defined in (4.2). Then

At the end of this section we investigate the set \({{\mathcal {U}}}_q^{(2^{\aleph _0})}\) which consists of all points having a continuum of *q*-expansions, and show that \({{\mathcal {U}}}_q^{(2^{\aleph _0})}\) has full Hausdorff measure.

### Lemma 5.5

### Proof

## 6 Examples and final remarks

In this section we consider some examples. The first example is an application of Theorems 1–3 to expansions with deleted digits set.

### Example 6.1

*q*-expansions with digits set \(\left\{ 0,1,3\right\} \). This is a special case of expansions with deleted digits (cf. [17]). Then

*k*different triadic expansions has the same Hausdorff dimension \(\log q_c/\log 3\) for any integer \(k\ge 1\). Moreover, by Theorem 3 it follows that \({{\mathcal {U}}}_3^{(\aleph _0)}\) is countably infinite, and

Theorem 1.1 gives a uniform formula for the Hausdorff dimension of \({{\mathcal {U}}}_q\) for \(q\in [q^*, \infty )\). Excluding the trivial case for \(q\in (1, q_c]\) that \({{\mathcal {U}}}_q=\left\{ 0, q/(q-1)\right\} \), it would be interesting to ask whether the Hausdorff dimension of \({{\mathcal {U}}}_q\) can be determined for \(q\in (q_c, q^*)\). In the following we give an example for which the Hausdorff dimension of \({{\mathcal {U}}}_q\) can be explicitly calculated.

### Example 6.2

*q*-expansion of \(q-1\) with alphabet \(\left\{ 0, q-1, q\right\} \) is \(q(q-1)^\infty \). Therefore, by Lemmas 3.1 and 3.2 of [24] it follows that \({{\mathcal {U}}}_q'\) is the set of sequences \((d_i)\in \left\{ 0,1,q\right\} ^\infty \) satisfying

*Question 1*. Can we give a uniform formula for the Hausdorff dimension of \({{\mathcal {U}}}_q\) for \(q\in (q_c, q^*)\)?

In beta expansions we know that the dimension function of the univoque set has a Devil’s staircase behavior (cf. [12]).

*Question 2*. Does the dimension function \(D(q):=\dim _H{{\mathcal {U}}}_q\) have a Devil’s staircase behavior in the interval \((q_c, q^*)\)?

By Theorem 3 one has that \({{\mathcal {U}}}_q^{(\aleph _0)}\) is countable for any \(q\in \mathcal {B}_2{\setminus }(q_c, q^*)\). Moreover, in Lemma 5.4 we show that \(\dim _H{{\mathcal {U}}}_q^{(\aleph _0)}\le \dim _H{{\mathcal {U}}}_q<1\) for any \(q\in (q_c, q^*)\). In view of Example 6.2 we ask the following

*Question 3*. Does there exist a \(q\in (q_c, q^*)\) such that \({{\mathcal {U}}}_q^{(\aleph _0)}\) has positive Hausdorff dimension?

## Notes

### Acknowledgements

The second author was supported by NSFC no. 11701302 and K. C. Wong Magna Fund at Ningbo University. The third author was supported by NSFC no. 11401516 and Jiangsu Province Natural Science Foundation for the Youth no. BK20130433. The forth author was supported by NSFC nos. 11271137, 11571144, 11671147 and in part by Science and Technology Commission of Shanghai Municipality (no. 18dz2271000)

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