Abstract
Given a positive integer M and a real number \(q >1\), a q -expansion of a real number x is a sequence \((c_i)=c_1c_2\ldots \) with \((c_i) \in \{0,\ldots ,M\}^\infty \) such that
It is well known that if \(q \in (1,M+1]\), then each \(x \in I_q:=\left[ 0,M/(q-1)\right] \) has a q-expansion. Let \(\mathcal {U}=\mathcal {U}(M)\) be the set of univoque bases \(q>1\) for which 1 has a unique q-expansion. The main object of this paper is to provide new characterizations of \(\mathcal {U}\) and to show that the Hausdorff dimension of the set of numbers \(x \in I_q\) with a unique q-expansion changes the most if q “crosses” a univoque base. Denote by \(\mathcal {B}_2=\mathcal {B}_2(M)\) the set of \(q \in (1,M+1]\) such that there exist numbers having precisely two distinct q-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (J Number Theory 129:741–754, 2009) and prove that
where \(q'=q'(M)\) is the Komornik–Loreti constant.
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1 Introduction
Non-integer base expansions have received much attention since the pioneering works of Rényi [25] and Parry [24]. Given a positive integer M and a real number \(q \in (1,M+1]\), a sequence \((d_i)=d_1d_2\ldots \) with digits \(d_i\in \left\{ 0,1,\ldots ,M\right\} \) is called a q -expansion of x or an expansion of x in base q if
It is well known that each \(x \in I_q:=[0,M/(q-1)]\) has a q-expansion. One such expansion—the greedy q -expansion—can be obtained by performing the so called greedy algorithm of Rényi which is defined recursively as follows: if \(d_1, \ldots ,d_{n-1}\) is already defined (no condition if \(n=1\)), then \(d_n\) is the largest element of \(\left\{ 0,\ldots ,M\right\} \) satisfying \(\sum _{i=1}^n d_i q^{-i} \le x\). Equivalently, \((d_i)\) is the greedy q-expansion of \(\sum _{i=1}^\infty d_i q^{-i}\) if and only if \(\sum _{i=n+1}^\infty d_i q^{-i+n}<1\) whenever \(d_n < M, n=1,2,\ldots \). Hence if \(1<q < r \le M+1\), then the greedy q-expansion of a number \(x \in I_q\) is also the greedy expansion in base r of a number in \(I_r\).
Let \({{\mathcal {U}}}_q\) be the univoque set consisting of numbers \(x\in I_q\) such that x has a unique q-expansion, and let \({{\mathcal {U}}}_q'\) be the set of corresponding expansions. Note that a sequence \((c_i)\) belongs to \({{\mathcal {U}}}_q'\) if and only if both the sequences \((c_i)\) and \((M-c_i):=(M-c_1)(M-c_2)\ldots \) are greedy q-expansions, hence \({{\mathcal {U}}}_q' \subseteq {{\mathcal {U}}}_r'\) whenever \(1< q < r \le M+1\). Many works are devoted to the univoque sets \({{\mathcal {U}}}_q\) (see, e.g., [10, 11, 14]). Recently, de Vries and Komornik investigated their topological properties in [8]. Komornik et al. considered their Hausdorff dimension in [19], and showed that the dimension function \(D: q\mapsto \dim _H{{\mathcal {U}}}_q\) behaves like a Devil’s staircase on \((1,M+1]\). For more information on the univoque set \({{\mathcal {U}}}_q\) we refer to the survey paper [15] and the references therein.
There is an intimate connection between the set \({{\mathcal {U}}}_q\) and the set of univoque bases \({{\mathcal {U}}}={{\mathcal {U}}}(M)\) consisting of numbers \(q>1\) such that 1 has a unique q-expansion over the alphabet \(\left\{ 0,1,\ldots ,M\right\} \). For instance, it was shown in [8] that \({{\mathcal {U}}}_q\) is closed if and only if q does not belong to the set \(\overline{{{\mathcal {U}}}}\). It is well-known that \({{\mathcal {U}}}\) is a Lebesgue null set of full Hausdorff dimension (cf. [6, 12, 19]). Moreover, the smallest element of \({{\mathcal {U}}}\) is the Komornik–Loreti constant (cf. [16, 17])
while the largest element of \({{\mathcal {U}}}\) is (of course) \(M+1\). Recently, Komornik and Loreti showed in [18] that its closure \(\overline{{{\mathcal {U}}}}\) is a Cantor set (see also, [9]), i.e., a nonempty closed set having neither isolated nor interior points. Writing the open set \((1,M+1] {\setminus } \overline{{{\mathcal {U}}}}=(1,M+1){\setminus } \overline{{{\mathcal {U}}}}\) as the disjoint union of its connected components, i.e.,
the left endpoints \(q_0\) in (1) run over the whole set \(\overline{{{\mathcal {U}}}}{\setminus }{{\mathcal {U}}}\), and the right endpoints \(q_0^*\) run through a subset of \({{\mathcal {U}}}\) (cf. [8]). Furthermore, each left endpoint \(q_0\) is algebraic, while each right endpoint \(q_0^*\in {{\mathcal {U}}}\) is transcendental (cf. [20]).
De Vries showed in [7], roughly speaking, that the sets \({{\mathcal {U}}}_q'\) change the most if we cross a univoque base. More precisely, it was shown that \(q \in {{\mathcal {U}}}\) if and only if \({{\mathcal {U}}}_r' {\setminus } {{\mathcal {U}}}_q'\) is uncountable for each \(r \in (q,M+1]\) and \(r \in \overline{{{\mathcal {U}}}}\) if and only if \({{\mathcal {U}}}_r' {\setminus } {{\mathcal {U}}}_q'\) is uncountable for each \(q \in (1,r)\).
The main object of this paper is to provide similar characterizations of \({{\mathcal {U}}}\) and \(\overline{{{\mathcal {U}}}}\) in terms of the Hausdorff dimension of the sets \({{\mathcal {U}}}_r' {\setminus } {{\mathcal {U}}}_q'\) after a natural projection. Furthermore, we characterize the sets \({{\mathcal {U}}}\) and \(\overline{{{\mathcal {U}}}}\) by looking at the Hausdorff dimensions of \({{\mathcal {U}}}\) and \(\overline{{{\mathcal {U}}}}\) locally.
Theorem 1.1
Let \(q\in (1, M+1]\). The following statements are equivalent.
-
(i)
\(q\in {{\mathcal {U}}}\).
-
(ii)
\(\dim _H\pi _{M+1}({{\mathcal {U}}}_r'{\setminus }{{\mathcal {U}}}_q')>0\) for any \(r\in (q, M+1]\).
-
(iii)
\(\dim _H{{\mathcal {U}}}\cap (q,r)>0\) for any \(r\in (q, M+1]\).
Theorem 1.2
Let \(q\in (1, M+1]\). The following statements are equivalent.
-
(i)
\(q\in \overline{{{\mathcal {U}}}}{\setminus }(\bigcup \left\{ q_0^*\right\} \cup \left\{ q'\right\} )\).
-
(ii)
\(\dim _H\pi _{M+1}({{\mathcal {U}}}_q'{\setminus }{{\mathcal {U}}}_p')>0\) for any \(p\in (1, q)\).
-
(iii)
\(\dim _H{{\mathcal {U}}}\cap (p, q)>0\) for any \(p\in (1, q)\).
It follows at once from Theorems 1.1 and 1.2 that \({{\mathcal {U}}}\) (or, equivalently, \(\overline{{{\mathcal {U}}}}\)) does not contain isolated points.
We remark that the projection map \(\pi _{M+1}\) in Theorem 1.1 (ii) can be replaced by \(\pi _\rho \) for any \(r\le \rho \le M+1\). Similarly, the projection map \(\pi _{M+1}\) in Theorem 1.2 (ii) can also be replaced by \(\pi _\rho \) with \(q\le \rho \le M+1\). We also point out that Theorems 1.1 and 1.2 strengthen the main result of [7] where the cardinality of the sets \({{\mathcal {U}}}_q' {\setminus } {{\mathcal {U}}}_p'\) with \(1< p < q \le M+1\) was determined.
Let \(\mathcal {B}_2\) be the set of bases \(q \in \left( 1,M+1\right] \) for which there exists a number \(x\in [0, M/(q-1)]\) having exactly two q-expansions. It was asked by Sidorov [26] whether \(\dim _H\mathcal {B}_2\cap (q', q'+\delta )>0 \) for any \(\delta >0\), where \(q'\) is the Komornik–Loreti constant. Since \({{\mathcal {U}}}\subseteq \mathcal {B}_2\) (see [26, Lemma 3.1]Footnote 1), Theorem 1.1 answers this question in the affirmative.
Corollary 1
\(\dim _H\mathcal {B}_2\cap (q', q'+\delta )>0\) for any \(\delta >0.\)
The rest of the paper is arranged as follows. In Sect. 2 we recall some properties of unique q-expansions. The proof of Theorems 1.1 and 1.2 will be given in Sect. 3.
2 Preliminaries
In this section we recall some properties of the univoque set \({{\mathcal {U}}}_q\). Throughout this paper, a sequence \((d_i)=d_1d_2\ldots \) is an element of \(\left\{ 0,\ldots ,M\right\} ^\infty \) with each digit \(d_i\) belonging to the alphabet \(\left\{ 0,\ldots ,M\right\} \). Moreover, for a word \({\mathbf {c}}=c_1\ldots c_n\) we mean a finite string of digits with each digit \(c_i\) from \(\left\{ 0,\ldots ,M\right\} \). For two words \({\mathbf {c}}=c_1\ldots c_n\) and \(\mathbf {d}=d_1\ldots d_m\) we denote by \({\mathbf {c}}\mathbf {d}=c_1\ldots c_nd_1\ldots d_m\) the concatenation of the two words. For an integer \(k\ge 1\) we denote by \({\mathbf {c}}^k\) the k-times concatenation of \({\mathbf {c}}\) with itself, and by \({\mathbf {c}}^\infty \) the infinite repetition of \({\mathbf {c}}\).
For a sequence \((d_i)\) we denote its reflection by \(\overline{(d_i)}:=(M-d_1)(M-d_2)\ldots \). Accordingly, for a word \({\mathbf {c}}=c_1\ldots c_n\) we denote its reflection by \(\overline{{\mathbf {c}}}:=(M-c_1)\ldots (M-c_n)\). If \(c_n<M\) we denote by \( {\mathbf {c}}^+:=c_1\ldots c_{n-1}(c_n+1). \) If \(c_n>0\) we write \({\mathbf {c}}^-:=c_1\ldots c_{n-1}(c_n-1)\).
We will use systematically the lexicographic ordering \(<, \le ,>\) and \(\ge \) between sequences and between words. For two sequences \((c_i), (d_i)\in \left\{ 0,1,\ldots , M\right\} ^\infty \) we say that \((c_i)<(d_i)\) 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 write \((c_i)\le (d_i)\) if \((c_i)<(d_i)\) or \((c_i)=(d_i)\). Similarly, we say \((c_i)>(d_i)\) if \((d_i)<(c_i)\), and \((c_i)\ge (d_i)\) if \((d_i)\le (c_i)\). We extend this definition to words in the obvious way. For example, for two words \({\mathbf {c}}\) and \(\mathbf {d}\) we write \({\mathbf {c}}<\mathbf {d}\) if \({\mathbf {c}}0^\infty <\mathbf {d}0^\infty \).
A sequence is called finite if it has a last nonzero element. Otherwise it is called infinite. So \(0^{\infty }:=00\ldots \) is considered to be infinite. For \(q\in (1,M+1]\) we denote by
the quasi-greedy q-expansion of 1 (cf. [5]), i.e., the lexicographically largest infinite q-expansion of 1. Let \(\beta (q)=(\beta _i(q))\) be the greedy q-expansion of 1 (cf. [24]), i.e., the lexicographically largest q-expansion of 1. For convenience, we set \(\alpha (1)=0^{\infty }\) and \(\beta (1)=10^{\infty }\), even though \(\alpha (1)\) is not a 1-expansion of 1.
Moreover, we endow the set \(\left\{ 0,\ldots ,M\right\} \) with the discrete topology and the set of all possible sequences \(\left\{ 0,1,\ldots ,M\right\} ^{\infty }\) with the Tychonoff product topology.
The following properties of \(\alpha (q)\) and \(\beta (q)\) were established in [24], see also [3].
Lemma 2.1
-
(i)
The map \(q\mapsto \alpha (q)\) is an increasing bijection from \([1, M+1]\) onto the set of all infinite sequences \((\alpha _i)\) satisfying
$$\begin{aligned} \alpha _{n+1}\alpha _{n+2}\ldots \le \alpha _1\alpha _2\ldots \quad \text {whenever}\quad \alpha _n<M. \end{aligned}$$ -
(ii)
The map \(q\mapsto \beta (q)\) is an increasing bijection from \([1, M+1]\) onto the set of all sequences \((\beta _i)\) satisfying
$$\begin{aligned} \beta _{n+1}\beta _{n+2}\ldots<\beta _1\beta _2\ldots \quad \text {whenever}\quad \beta _n<M. \end{aligned}$$
Lemma 2.2
-
(i)
\(\beta (q)\) is infinite if and only if \(\beta (q)=\alpha (q)\).
-
(ii)
If \(\beta (q)=\beta _1\ldots \beta _m 0^\infty \) with \(\beta _m>0\), then \(\alpha (q)=(\beta _1\ldots \beta _m^-)^\infty .\)
-
(iii)
The map \(q\mapsto \alpha (q)\) is left-continuous, while the map \(q\mapsto \beta (q)\) is right-continuous.
In order to investigate the unique expansions we need the following lexicographic characterization of \({{\mathcal {U}}}_q'\) (cf. [3]).
Lemma 2.3
Let \(q\in (1,M+1]\). Then \((d_i)\in {{\mathcal {U}}}_q'\) if and only if
Note that \(q\in {{\mathcal {U}}}\) if and only if \(\alpha (q)\) is the unique q-expansion of 1. Then Lemma 2.3 yields a characterization of \({{\mathcal {U}}}\) (see also, [11, 17]).
Lemma 2.4
Let \(q \in (1,M+1)\). Then \(q\in {{\mathcal {U}}}\) if and only if \(\alpha (q)=(\alpha _i(q))\) satisfies
Consider a connected component \((q_0, q_0^*)\) of \((q', M+1){\setminus }\overline{{{\mathcal {U}}}}\) as in (1). Then there exists a (unique) word \({{\mathbf {t}}}=t_1\ldots t_p\) such that (cf. [8, 20])
where \(g^n=\underbrace{g\circ \cdots \circ g}_n\) denotes the n-fold composition of g with itself, and
We point out that the word \({{\mathbf {t}}}=t_1\ldots t_p\) in the definitions of \(\alpha (q_0)\) and \(\alpha (q_0^*)\) is called an admissible block in [20, Definition 2.1] which satisfies the following lexicographical inequalities: \(t_p<M\) and for any \(1\le i\le p\) we have
We also mention that the limit \(\lim _{n\rightarrow \infty }g^n({{\mathbf {t}}})\) stands for the infinite sequence beginning with \( {{\mathbf {t}}}^+\overline{{{\mathbf {t}}}}\,\overline{{{\mathbf {t}}}^+}{{\mathbf {t}}}^+\;\overline{{{\mathbf {t}}}^+}{{\mathbf {t}}}\,{{\mathbf {t}}}^+\overline{{{\mathbf {t}}}}\ldots , \) and the existence of this limit was shown by Allouche [2].
In this case \((q_0, q_0^*)\) is called the connected component generated by \({{\mathbf {t}}}\). The closed interval \([q_0, q_0^*]\) is the so called admissible interval generated by \({{\mathbf {t}}}\) (see [20, Definition 2.4]). Furthermore, the sequence
is a generalized Thue–Morse sequence (cf. [20, Definition 2.2], see also [1]).
The following lemma for the generalized Thue–Morse sequence \(\alpha (q_0^*)\) was established in [20, Lemma 4.2].
Lemma 2.5
Let \((q_0, q_0^*)\subset (q', M+1){\setminus }\overline{{{\mathcal {U}}}}\) be a connected component generated by \(t_1\ldots t_p\). Then the sequence \((\theta _i)=\alpha (q_0^*)\) satisfies
for any \(n\ge 0\) and any \(0\le i<2^n p\).
Finally, we recall some topological properties of \({{\mathcal {U}}}\) and \(\overline{{{\mathcal {U}}}}\) which were essentially established in [8, 18] (see also, [9]).
Lemma 2.6
-
(i)
If \(q\in {{\mathcal {U}}}\), then there exists a decreasing sequence \((r_n)\) of elements in \(\bigcup \left\{ q_0^*\right\} \) that converges to q as \(n\rightarrow \infty \);
-
(ii)
If \(q\in \overline{{{\mathcal {U}}}}{\setminus }(\bigcup \left\{ q_0^*\right\} \cup \left\{ q'\right\} )\), then there exists an increasing sequence \((p_n)\) of elements in \(\bigcup \left\{ q_0^*\right\} \) that converges to q as \(n\rightarrow \infty \).
We remark here that the bases \(q_0^*\) are called de Vries–Komornik numbers which were shown to be transcendental in [20]. By Lemma 2.6 it follows that the set of de Vries–Komornik numbers is dense in \(\overline{{{\mathcal {U}}}}\).
3 Proofs of Theorems 1.1 and 1.2
3.1 Proof of Theorem 1.1 for (i) \(\Leftrightarrow \) (ii).
For each connected component \((q_0, q_0^*)\) of \((q', M+1){\setminus }\overline{{{\mathcal {U}}}}\) we construct a sequence of bases \((r_n)\) in \({{\mathcal {U}}}\) strictly decreasing to \(q_0^*\).
Lemma 3.1
Let \((q_0, q_0^*)\subset (q', M+1){\setminus }\overline{{{\mathcal {U}}}}\) be a connected component generated by \(t_1\ldots t_p\), and let \((\theta _i)=\alpha (q_0^*)\). Then for each \(n\ge 1\), the number \(r_n\in {{\mathcal {U}}}\) determined by
belongs to \({{\mathcal {U}}}\). Furthermore, \((r_n)\) is a strictly decreasing sequence that converges to \(q_0^*\).
Proof
Using (2) one may verify that the sequence \((\theta _i)\) satisfies
for all \(n\ge 0\). Now fix \(n\ge 1\). We claim that
for all \(i\ge 1\), where \(\sigma \) is the left shift on \(\left\{ 0,\ldots , M\right\} ^\infty \) defined by \(\sigma ((c_i))=(c_{i+1})\). By periodicity it suffices to prove (3) for \(0<i<2^{n+1}p\). We distinguish between the following three cases: (I) \(0< i<2^np\); (II) \(i=2^n p\); (III) \(2^n p<i<2^{n+1}p\).
Case (I). \(0< i<2^n p\). Then by Lemma 2.5 it follows that
and
This implies (3) for \(0<i<2^n p\).
Case (II). \(i=2^n p\). Note by [17] that \(\alpha _1(q')=[M/2]+1\) (see also, [4]), where [y] denotes the integer part of a real number y. Then by using \(q_0^*>q'\) in Lemma 2.1 we have
This, together with \(n\ge 1\), implies
So, (3) holds true for \(i=2^n p\).
Case (III). \(2^n p< i<2^{n+1}p\). Write \(j=i-2^n p\). Then \(0< j<2^n p\). Once again, we infer from Lemma 2.5 that
and
This yields (3) for \(2^n p<i<2^{n+1}p\).
Note by Lemma 2.5 that
for any \(i\ge 0\). Then by (3) and Lemma 2.4 it follows that there exists \(r_n\in {{\mathcal {U}}}\) such that
In the following we prove \(r_n\searrow q_0^*\) as \(n\rightarrow \infty \). For \(n\ge 1\) we observe that
Then by Lemma 2.1 (ii) we have \(r_{n+1}<r_{n}\). Note that \(\beta (q_0^*)=\alpha (q_0^*)=(\theta _i)\), and
Hence, we conclude from Lemma 2.2 (iii) that \(r_n\searrow q_0^*\) as \(n\rightarrow \infty \). \(\square \)
Lemma 3.2
Let \((q_0, q_0^*)\subset (q',M+1){\setminus }\overline{{{\mathcal {U}}}}\) be a connected component generated by \(t_1\ldots t_p\), and let \((\theta _i)=\alpha (q_0^*)\). Then for any \(n\ge 1\) and any \(0\le i< 2^{n}p\) we have
and thus (by symmetry),
where \(\xi _n:=\theta _1\ldots \theta _{2^n p}\).
Proof
By symmetry it suffices to prove (4).
Note that \( \xi _n\overline{\xi _n}=\theta _1\ldots \theta _{2^{n+1}p}^-\) and \(\xi _n\overline{\xi _n^-}=\theta _1\ldots \theta _{2^{n+1}p}.\) Then by Lemma 2.5 it follows that
and
for any \(0\le i<2^{n}p\).
So, it suffices to prove the inequalities
for any \(0\le i<2^{n}p\). By Lemma 2.5 it follows that for any \(0\le i<2^n p\) we have
and
This proves (5). \(\square \)
Lemma 3.3
Let \((q_0, q_0^*)\subset (q',M+1){\setminus }\overline{{{\mathcal {U}}}}\) be a connected component generated by \(t_1\ldots t_p\). Then \( \dim _H\pi _{M+1}({{\mathcal {U}}}_r'{\setminus }{{\mathcal {U}}}_{q_0^*}')>0 \) for any \(r\in (q_0^*, M+1]\).
Proof
Take \(r\in (q_0^*, M+1]\). By Lemma 3.1 there exists \(n\ge 1\) such that
Write \((\theta _i)=\alpha (q_0^*)\) and let \(\xi _n=\theta _1\ldots \theta _{2^n p}\). Denote by \(X_A^{(n)}\) the subshift of finite type over the states \(\left\{ \xi _n, \xi _n^-, \overline{\xi _n}, \overline{\xi _n^-}\right\} \) with adjacency matrix
Note that \(\alpha (r_n)=\theta _1\ldots \theta _{2^n p}(\theta _{2^n p+1}\ldots \theta _{2^{n+1}p})^\infty \). Then by Lemmas 3.2 and 2.3 it follows that
Furthermore, note that
Then by Lemmas 2.3 and 3.1 it follows that any sequence starting at
can not belong to \({{\mathcal {U}}}_{r_{n+2}}'\). Therefore, by (6) we obtain
Note that the subshift of finite type \(X_A^{(n)}\) is irreducible (cf. [22]), and the image \(\pi _{M+1}(X_A^{(n)})\) is a graph-directed set satisfying the open set condition (cf. [23]). Then by (7) it follows that
\(\square \)
The following lemma can be shown in a way which resembles closely the analysis in [21, pp. 2829–2830]. For the sake of completeness we include a sketch of its proof.
Lemma 3.4
Let \((q_0, q_0^*)\subset (q',M+1){\setminus }\overline{{{\mathcal {U}}}}\) be a connected component. Then \(\dim _H\pi _{M+1}({{\mathcal {U}}}_{q_0^*}'{\setminus }{{\mathcal {U}}}_{q_0}')=0\).
Proof
(Sketch of the proof) Suppose that \((q_0, q_0^*)\) is a connected component generated by \({{\mathbf {t}}}=t_1\ldots t_p\). Then
where \(g(\cdot )\) is defined in (2).
For \(n\ge 0\) let \({\omega }_n:=g^n({{\mathbf {t}}})^+\). Take \((d_i)\in {{\mathcal {U}}}_{q_0^*}'{\setminus }{{\mathcal {U}}}_{q_0}'\). Then by using (8) and Lemma 2.3 it follows that there exists \(m\ge 1\) such that
or symmetrically,
Suppose \((d_{m+i})\ne {{\mathbf {t}}}^\infty \) and \((d_{m+i})\ne \overline{{{\mathbf {t}}}^\infty }\). Then there exists \(u\ge m\) such that
-
If \(d_{u+1}\ldots d_{u+p}={\omega }_0={{\mathbf {t}}}^+\), then by (9) and Lemma 2.3 it follows that
$$\begin{aligned} d_{u+p+1}\ldots d_{u+2p}=\overline{{{\mathbf {t}}}^+} \quad \text {or}\quad d_{u+p+1}\ldots d_{u+2p}=\overline{{{\mathbf {t}}}}. \end{aligned}$$This implies \(d_{u+1}\ldots d_{u+2p}={{\mathbf {t}}}^+\overline{{{\mathbf {t}}}^+} ={\omega }_0\,\overline{{\omega }_0}\) or \( d_{u+1}\ldots d_{u+2p}={{\mathbf {t}}}^+\overline{{{\mathbf {t}}}}={\omega }_1.\)
-
If \(d_{u+1}\ldots d_{u+p}=\overline{{\omega }_0}=\overline{{{\mathbf {t}}}^+}\), then by (10) and Lemma 2.3 it follows that
$$\begin{aligned} d_{u+p+1}\ldots d_{u+2p}={{{\mathbf {t}}}^+} \quad \text {or}\quad d_{u+p+1}\ldots d_{u+2p}={{\mathbf {t}}}. \end{aligned}$$This yields that \(d_{u+1}\ldots d_{u+2p}=\overline{{\omega }_0}\,{\omega }_0\) or \(d_{u+1}\ldots d_{u+2p}=\overline{{\omega }_1}\).
Note that for each \(n\ge 0\) the word \(g^n({{\mathbf {t}}})^+\,\overline{g^n({{\mathbf {t}}})}\) is a prefix of \(\alpha (q_0^*)\). By iteration of the above arguments, one can show that if \(d_{v+1}\ldots d_{v+2^n p}={\omega }_n\), then \(d_{v+1}\ldots d_{v+2^{n+1}p}={\omega }_n\overline{{\omega }_n}\) or \({\omega }_{n+1}\). Symmetrically, if \(d_{v+1}\ldots d_{v+2^n p}=\overline{{\omega }_n}\), then \(d_{v+1}\ldots d_{v+2^{n+1}p}=\overline{{\omega }_n}{\omega }_n\) or \(\overline{{\omega }_{n+1}}\).
Hence, we conclude that \((d_{i})\) must end with
or its reflections, where \(s_n\in \left\{ 0,1\right\} \) and
Here \(*\) is an element of the set \(\left\{ 0,1,2,\ldots \right\} \cup \left\{ \infty \right\} \).
Since the length of \({\omega }_n=g^n({{\mathbf {t}}})^+\) grows exponentially fast as \(n\rightarrow \infty \), we conclude that \(\dim _H\pi _{M+1}({{\mathcal {U}}}_{q_0^*}'{\setminus }{{\mathcal {U}}}_{q_0}')=0\). \(\square \)
Proof of Theorem 1.1 for (i) \(\Leftrightarrow \) (ii) First we prove (i) \(\Rightarrow \) (ii). If \(q=q_0^*\) is the right endpoint of a connected component of \((q', M+1){\setminus }\overline{{{\mathcal {U}}}}\), then by Lemma 3.3 we have
Clearly, it is trivial when \(q=M+1\). Now we take \(q\in ({{\mathcal {U}}}{\setminus }\left\{ M+1\right\} ){\setminus }\bigcup \left\{ q_0^*\right\} \) and take \(r\in (q, M+1]\). By Lemma 2.6 (i) one can find \(q_0^*\in (q,r)\), and therefore by Lemma 3.3 we obtain
Now we prove (ii) \(\Rightarrow \) (i). Take \(q\in (1, M+1]{\setminus }{{\mathcal {U}}}\). We will show that \(\dim _H\pi _{M+1}({{\mathcal {U}}}_r'{\setminus }{{\mathcal {U}}}_q')=0\) for some \(r\in (q, M+1]\). Note that \(\bigcup \left\{ q_0\right\} =\overline{{{\mathcal {U}}}}{\setminus }{{\mathcal {U}}}\). Then by (1) it follows that
Therefore, it suffices to prove \(\dim _H\pi _{M+1}({{\mathcal {U}}}_r'{\setminus }{{\mathcal {U}}}_q')=0\) for some \(r\in (q, M+1]\). We distinct the following two cases.
Case (I). \(q\in (1,q')\). Then for any \(r\in (q,q')\) we have
where the last equality follows by [21, Theorem 4.6] (see also [4, 14]).
Case (II). \(q\in [q_0, q_0^*)\). Then for any \(r\in (q,q_0^*)\) we have by Lemma 3.4 that
\(\square \)
3.2 Proof of Theorem 1.1 for (i) \(\Leftrightarrow \) (iii)
The following property for the Hausdorff dimension is well-known (cf. [13, Proposition 2.3]).
Lemma 3.5
Let \(f: (X, d_1)\rightarrow (Y, d_2)\) be a map between two metric spaces . If there exist constants \(C>0\) and \(\lambda >0\) such that
for any \(x, y\in X\), then \(\dim _H X\ge \lambda \dim _H f(X)\).
Lemma 3.6
Let \(q\in {{{\mathcal {U}}}}{\setminus }\left\{ M+1\right\} \). Then for any \(r\in (q, M+1)\) we have
Proof
Fix \(q\in {{{\mathcal {U}}}}{\setminus }\left\{ M+1\right\} \) and \(r\in (q, M+1)\). Then Lemma 2.6 yields that \({{\mathcal {U}}}\cap (q, r)\) contains infinitely many elements. Take \(p_1, p_2\in {{\mathcal {U}}}\cap (q, r)\) with \(p_1<p_2\). Then by Lemma 2.1 we have \(\alpha (p_1)<\alpha (p_2)\). So, there exists \(n\ge 1\) such that
This implies
Note that \(r<M+1\). By Lemma 2.1 we have \(\alpha (r)<\alpha (M+1)=M^\infty \). Then there exists \(N\ge 1\) such that
Therefore, by (11) and Lemma 2.3 we obtain
Note that \(p_1, p_2\) are elements of \({{\mathcal {U}}}\). Then \(p_2>p_1\ge q'\). This implies
Therefore, by (12) it follows that
Furthermore, by Lemma 2.1 it follows that \(\pi _{M+1}(\alpha (p_2))-\pi _{M+1}(\alpha (p_1))\ge 0\). Hence, by using
in Lemma 3.5 we establish the lemma. \(\square \)
Lemma 3.7
Let \((q_0, q_0^*)\) be a connected component of \((q', M+1){\setminus }\overline{{{\mathcal {U}}}}\). Then \(\dim _H{{\mathcal {U}}}\cap (q_0^*, r)>0\) for any \(r\in (q_0^*, M+1]\).
Proof
Suppose that \((q_0, q_0^*)\) is a connected component generated by \(t_1\ldots t_p\). Let \((\theta _i)=\alpha (q_0^*)\). For \(n\ge 2\) we write \(\xi _n=\theta _1\ldots \theta _{2^n p}\), and denote by
Here \(X_A^{(n)}(\overline{\xi _n})\) is the follower set of \(\overline{\xi _n}\) in the subshift of finite type \(X_A^{(n)}\) defined in (7). Now we claim that any sequence \((d_i)\in \varGamma _n'\) satisfies
Take \((d_i)\in \varGamma _n'\). Then we deduce by the definition of \(\varGamma _n'\) that
We will split the proof of (13) into the following five cases.
-
(a)
\(1\le j<2^{n-1}p\). By (14) and Lemma 2.5 it follows that
$$\begin{aligned} \overline{\theta _1\ldots \theta _{2^{n-1}p-j}}<d_{j+1}\ldots d_{2^{n-1}p}=\theta _{j+1}\ldots \theta _{2^{n-1}p}\le \theta _1\ldots \theta _{2^{n-1}p-j}, \end{aligned}$$and
$$\begin{aligned} d_{2^{n-1}p+1}\ldots d_{2^{n-1}p+j}=\overline{\theta _1\ldots \theta _j}<\theta _{2^{n-1}p-j+1}\ldots \theta _{2^{n-1}p}. \end{aligned}$$This implies that (13) holds for all \(1\le j<2^{n-1}p\).
-
(b)
\(2^{n-1}p\le j<2^n p\). Let \(k=j-2^{n-1}p\). Then \(0\le k<2^{n-1}p\). Clearly, if \(k=0\), then by using \(\theta _1>\overline{\theta _1}\) and \(n\ge 2\) it yields that
$$\begin{aligned} \overline{\theta _1\ldots \theta _{2^{n-1}p}}<d_{j+1}\ldots d_{2^n p}=\overline{\theta _1\ldots \theta _{2^{n-1}p}}\,^+<\theta _1\ldots \theta _{2^{n-1}p}. \end{aligned}$$Now we assume \(1\le k<2^{n-1}p\). Then by (14) and Lemma 2.5 it follows that
$$\begin{aligned} \overline{\theta _1\ldots \theta _{2^{n-1}p-k}}<d_{j+1}\ldots d_{2^n p}=\overline{\theta _{k+1}\ldots \theta _{2^{n-1}p}}\,^+\le \theta _1\ldots \theta _{2^{n-1}p-k}, \end{aligned}$$and
$$\begin{aligned} d_{2^{n}p+1}\ldots d_{2^n p+k}=\overline{\theta _1\ldots \theta _k}<\theta _{2^{n-1}p-k+1}\ldots \theta _{2^{n-1}p}. \end{aligned}$$Therefore, (13) holds for all \(2^{n-1}p\le j<2^n p\).
-
(c)
\(2^n p\le j<2^n p+2^{n-1}p\). Let \(k=j-2^n p\). Then in a similar way as in Case (b) one can prove (13).
-
(d)
\(2^np+2^{n-1}p\le j<2^{n+1}p\). Let \(k=j-2^np-2^{n-1}p\). Again by the same arguments as in Case (b) we obtain (13).
-
(e)
\(j\ge 2^{n+1}p\). Note that
$$\begin{aligned} d_1\ldots d_{2^{n+1}p}=\theta _1\ldots \theta _{2^{n-1}p}\left( \overline{\theta _1\ldots \theta _{2^{n-1}p}}\,^+\right) ^3>\theta _1\ldots \theta _{2^{n+1}p}. \end{aligned}$$
Therefore, by (13) and Lemma 2.4 it follows that any sequence in \(\varGamma _n'\) corresponds to a unique base \(q\in {{\mathcal {U}}}\). Furthermore, by (14) and Lemma 3.1 each sequence \((d_i)\in \varGamma _n'\) satisfies
Then by Lemma 2.1 it follows that
Fix \(r>q_0^*\). So by Lemma 3.1 there exists a sufficiently large integer \(n\ge 2\) such that
Note by the proof of Lemma 3.3 that \(X_A^{(n)}\) is an irreducible subshift of finite type over the states \(\left\{ \xi _n, \xi _n^-, \overline{\xi _n}, \overline{\xi _n^-}\right\} \). Hence, by (15) and Lemma 3.6 it follows that
\(\square \)
Proof of Theorem 1.1 for (i) \(\Leftrightarrow \) (iii) First we prove (i) \(\Rightarrow \) (iii). Excluding the trivial case \(q=M+1\) we take \(q\in {{\mathcal {U}}}{\setminus }\left\{ M+1\right\} \). Suppose that \(r \in (q,M+1]\). If \(q=q_0^*\), then by Lemma 3.7 we have \( \dim _H{{\mathcal {U}}}\cap (q, r)>0. \)
If \(q\in ({{\mathcal {U}}}{\setminus }\left\{ M+1\right\} ) {\setminus }\bigcup \left\{ q_0^*\right\} \), then by Lemma 2.6 (i) there exists \(q_0^*\in (q, r)\). So, by Lemma 3.7 we have
Now we prove (iii) \(\Rightarrow \) (i). Suppose on the contrary that \(q\in (1,M+1]{\setminus }{{\mathcal {U}}}\). We will show that \({{\mathcal {U}}}\cap (q,r)=\emptyset \) for some \(r\in (q, M+1]\). Take \(q\in (1, M+1]{\setminus }{{\mathcal {U}}}\). By (1) it follows that
This implies that \({{\mathcal {U}}}\cap (q,r)=\emptyset \) for \(r \in (q, M+1]\) sufficiently close to q. \(\square \)
3.3 Proof of Theorem 1.2
Proof of Theorem 1.2 (i) \(\Rightarrow \) (ii) Take \(q\in \overline{{{\mathcal {U}}}}{\setminus }(\bigcup \left\{ q_0^*\right\} \cup \left\{ q'\right\} )\) and \(p\in (1,q)\). By Lemma 2.6 (ii) there exists \(q_0^*\in (p,q)\). Hence, by Lemma 3.3 it follows that
(ii) \(\Rightarrow \) (i). Suppose on the contrary that \(q\notin \overline{{{\mathcal {U}}}}{\setminus }(\bigcup \left\{ q_0^*\right\} \cup \left\{ q'\right\} )\). Then by (1) we have
By using Lemma 3.4 it follows that for \(p\in (1,q)\) sufficiently close to q we have \(\dim _H\pi _{M+1}({{\mathcal {U}}}_q'{\setminus }{{\mathcal {U}}}_p')=0\).
(i) \(\Rightarrow \) (iii). Take \(q\in \overline{{{\mathcal {U}}}}{\setminus }(\bigcup \left\{ q_0^*\right\} \cup \left\{ q'\right\} )\) and \(p\in (1,q)\). By Lemma 2.6 (ii) there exists \(q_0^*\in (p,q)\). Hence, by Lemma 3.7 it follows that
(iii) \(\Rightarrow \) (i). Suppose \(q\notin \overline{{{\mathcal {U}}}}{\setminus }(\bigcup \left\{ q_0^*\right\} \cup \left\{ q'\right\} )\). Then by (1) we have \( q\in (1,q']\cup \bigcup (q_0, q_0^*]. \) So, for \(p\in (1, q)\) sufficiently close to q we have \({{\mathcal {U}}}\cap (p,q)=\emptyset \). \(\square \)
Notes
This also follows directly from the observation that \(q^{-1}\) has exactly two q-expansions whenever \(q \in {{\mathcal {U}}}\).
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Acknowledgements
The authors thank the anonymous referees for many useful comments. Derong Kong was supported by NSFC No. 11401516 and Jiangsu Province Natural Science Foundation for the Youth No. BK20130433. Wenxia Li was supported by NSFC Nos. 11271137, 11571144, 11671147 and Science and Technology Commission of Shanghai Municipality (STCSM) No. 13dz2260400. Fan Lü was supported by NSFC No. 11601358.
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Kong, D., Li, W., Lü, F. et al. Univoque bases and Hausdorff dimension. Monatsh Math 184, 443–458 (2017). https://doi.org/10.1007/s00605-017-1047-9
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DOI: https://doi.org/10.1007/s00605-017-1047-9