1 Introduction
Let $T:[0,1)\to [0,1)$ be the Gauss map defined by
Every irrational number $x\in [0,1)$ can be uniquely expanded into an infinite form
where $a_{n}(x)=\lfloor 1/T^{n-1}(x)\rfloor $ are called the partial quotients of $x.$ (Here $\lfloor \cdot \rfloor $ denotes the greatest integer less than or equal to a real number and $T^0$ denotes the identity map.) For simplicity of notation, we write (1.1) as
It is clear that the Gauss transformation T acts as the shift map on the continued fraction system. That is, for each $x=[a_1(x),a_2(x),a_3(x),\ldots ]\in [0,1)\cap \mathbb {Q}^{c},$
Gauss observed that T is measure-preserving and ergodic with respect to the Gauss measure $\mu $ defined by
For more information on the continued fraction expansion, see [Reference Khintchine3].
The metrical theory of continued fractions, which concerns the properties of the partial quotients for almost all $x\in [0,1),$ is one of the major themes in the study of continued fractions. Wang and Wu [Reference Wang and Wu7] considered the metrical properties of the maximal run-length function
which counts the longest run of the same symbol among the first n partial quotients of x. They proved that, for $\mu $ almost all $x\in [0,1),$
Song and Zhou [Reference Song and Zhou6] gave a more subtle characterisation of the function $R_{n}(x).$ In this paper, we continue the study by considering the shortest distance function
This is motivated by the behaviour of the shortest distance between two orbits,
in the continued fraction system. Shi et al. [Reference Shi, Tan and Zhou5] proved that, for $\mu ^{2}$ almost all $(x,y)\in [0,1)\times [0,1),$
where $H_{2}$ is the Rényi entropy defined by (1.2). Investigating the shortest distance between two orbits amounts to estimating the longest common substrings between two sequences of partial quotients. In fact, [Reference Shi, Tan and Zhou5] focused on the asymptotics of the length of the longest common substrings in two sequences of partial quotients.
For $n\geq 1$ and $(a_{1},\ldots ,a_{n})\in \mathbb {N}^{n}$ , we call
an nth cylinder. For $m\geq 2$ , we define the generalised Rényi entropy with respect to the Gauss measure $\mu $ by
The existence of the limit (1.2) for the Gauss measure $\mu $ was established in [Reference Haydn and Vaienti2].
Theorem 1.1. For $\mu ^{m}$ -almost all $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$
Here we use the convention that ${1}/{0}=\infty $ and ${1}/{\infty }=0.$
It is natural to study the exceptional set in this limit theorem. We define the exceptional set as
and the level set as
Throughout the paper, $\dim _{H}A$ denotes the Hausdorff dimension of the set $A.$
Theorem 1.2. For any $\alpha $ with $0\leq \alpha \leq \infty ,$ $\dim _{H}\widetilde {E}=\dim _{H}E(\alpha )=m.$
In fact, Theorem 1.2 follows immediately from the following more general result. For any $0\leq \alpha \leq \beta \leq \infty ,$ set
Theorem 1.3. For any $\alpha ,\beta $ with $0\leq \alpha \leq \beta \leq \infty ,$ $\dim _{H}E(\alpha ,\beta )=m.$
2 Preliminaries
In this section, we fix some notation and recall some basic properties of continued fraction expansions. A detailed account of continued fractions can be found in Khintchine’s book [Reference Khintchine3].
For any irrational number $x\in [0,1)$ with continued fraction expansion (1.1), we denote by
the nth convergent of $x.$ With the conventions
we have, for any $n\ge 0,$
Obviously, $q_{n}(x)$ is determined by the first n partial quotients $a_{1}(x),\ldots ,a_{n}(x).$ So we also write $q_{n}(a_{1}(x),\ldots ,a_{n}(x))$ in place of $q_{n}(x).$ If no confusion is likely to arise, we write $a_{n}$ and $q_{n}$ in place of $a_{n}(x)$ and $q_{n}(x),$ respectively.
Proposition 2.1 [Reference Khintchine3].
For $n\geq 1$ and $(a_{1},\ldots ,a_{n})\in \mathbb {N}^{n}$ :
-
(1) $ q_n\ge 2^{(n-1)/2}$ and
$$ \begin{align*} \prod^{n}_{k=1}a_{k}\leq q_n \leq \prod^{n}_{k=1}(a_{k}+1)\leq 2^{n}\prod^{n}_{k=1}a_{k};\end{align*} $$ -
(2) the length of $I_{n}(a_{1},\ldots ,a_{n})$ satisfies
$$ \begin{align*} \frac{1}{2q_n^2} \leq |I_{n}(a_{1},\ldots,a_{n})|=\frac{1}{(q_n+q_{n-1})q_n} \leq \frac{1}{q_n^2}. \end{align*} $$
The following $\psi $ -mixing property is essential in proving Theorem 1.1.
Lemma 2.2 [Reference Philipp4].
For any $k\geq 1,$ let $\mathbb {B}^{k}_{1}=\sigma (a_{1},\ldots ,a_{k})$ and let $\mathbb {B}^{\infty }_{k}=\sigma (a_{k},a_{k+1},\ldots )$ denote the $\sigma $ -algebras generated by the random variables $(a_{1},\ldots ,a_{k})$ and $(a_{k},a_{k+1},\ldots )$ respectively. Then, for any $E\in \mathbb {B}^{k}_{1}$ and $F\in \mathbb {B}^{\infty }_{k+n},$
where $|\theta |\leq K, \rho <1$ and $K, \rho $ are positive constants independent of $E, F, n$ and $k.$
To estimate the measure of a limsup set in a probability space, the following lemma is widely used.
Lemma 2.3 (Borel–Cantelli lemma).
Let $(\Omega , \mathcal {B}, \nu )$ be a finite measure space and let $\{A_n\}_{n\ge 1}$ be a sequence of measurable sets. Define $A=\bigcap _{N=1}^{\infty }\bigcup _{n=N}^{\infty }A_n.$ Then
Let $\mathbb {K}=\{k_{n}\}_{n\geq 1}$ be a subsequence of $\mathbb {N}$ that is not cofinite. Define a mapping $\phi _{\mathbb {K}}:[0,1)\cap \mathbb {Q}^{c}\rightarrow [0,1)\cap \mathbb {Q}^{c}$ as follows. For each $x=[a_1,a_2,\ldots ]\in [0,1)\cap \mathbb {Q}^{c},$ put $\phi _{\mathbb {K}}(x)=\overline {x}=[c_1,c_2,\ldots ],$ where $[c_1,c_2,\ldots ]$ is obtained by eliminating all the terms $a_{k_{n}}$ from the sequence $a_1, a_2,\ldots .$ Let $\{b_{n}\}_{n\geq 1}$ be a sequence with $b_{n}\in \mathbb {N},n\geq 1.$ Write
Lemma 2.4 [Reference Song and Zhou6].
Assume that $\{b_{n}\}_{n\geq 1}$ is bounded. If the sequence $\mathbb {K}$ is of density zero in $\mathbb {N},$ that is,
where $\sharp $ denotes the number of elements in a set, then
We close this section by citing Marstrand’s product theorem.
Lemma 2.5 [Reference Falconer1].
If $E,F\subset \mathbb {R}^{d}$ for some $d,$ then $\dim _{H}(E\times F)\geq \dim _{H}E+\dim _{H}F.$
3 Proof of Theorem 1.1
Theorem 1.1 can be proved from the following two propositions.
Proposition 3.1. For $\mu ^{m}$ -almost all $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$
Proof. We can assume that $H_{m}>0$ (the case $H_m=0$ is obvious). Fix $s_1< s_2< (m-1)H_{m}.$ By the definition of the $H_{m},$
for sufficiently large n. Set $u_{n}=\lfloor {\log n}/{s_1}\rfloor .$ Note that, for any $(x_{1},\ldots ,x_{m})\in [0,1)^{m}$ with $M_{n,m}(x_{1},\ldots ,x_{m})=k$ , there exists i with $0\leq i\leq n-k$ such that
for $j=1,\ldots ,k.$ We deduce
By the invariance of $\mu $ under $T,$ it follows that
where $C=\sum _{k=1}^{\infty }\exp \{-{(s_1+s_2)k}/{2}\}.$ Choose an infinite subsequence of integers $\{n_{k}\}_{k\geq 1},$ where $n_{k}=k^{L}$ and $L\cdot {(s_2-s_1)}/{2s_1}>1.$ Then
From the Borel–Cantelli Lemma 2.3, for almost all $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$
for sufficiently large k. Thus,
Therefore, by the arbitrariness of $s_{1},$
This completes the proof.
Proposition 3.2. For $\mu ^{m}$ -almost all $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$
Proof. We can assume that $H_{m}<\infty $ (the case $H_m=\infty $ is obvious). For $1\le d<n$ , set
We denote $\{(x_{1},\ldots ,x_{m})\in [0,1)^{m}: M_{n,m}(x_{1},\ldots ,x_{m})<k\}$ by $\{M_{n,m}<k\}$ for brevity.
For any $s>(m-1)H_{m},$ by the definition of the $H_{m},$
for sufficiently large n. Let $u_{n}=\lfloor {\log n}/{s}\rfloor $ and $l_{n}=\lfloor {n}/{u_{n}^{2}}\rfloor .$ Then
By Lemma 2.2, it follows that
where the penultimate inequality follows from (3.2) and the two facts $ (1-x)<\exp (-x)$ for $0<x<1$ and $\lim _{n\to \infty }(1+1/n)^{n}=e$ , and the last inequality follows because $\theta \rho ^{u_{n}^{2}-u_{n}}{m\cdot n}/{u_{n}^{2}}\rightarrow 0$ as $n\rightarrow \infty .$ Thus,
From the Borel–Cantelli Lemma, for $\mu ^{m}$ -almost all $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$
This completes the proof of Proposition 3.2 and of Theorem 1.1.
4 Proof of Theorem 1.3
This section is devoted to the proof of Theorem 1.3. Our strategy is to construct Cantor-like subsets with full Hausdorff dimension. The proof is divided into several cases according to the values of $\alpha $ and $\beta .$ We give a detailed proof for the case $0<\alpha <\beta <\infty $ and a sketch of the proof for the remaining cases.
Case 1: $0<\alpha <\beta <\infty .$
Choose two positive integer sequences $\{n_{k}\}_{k\geq 1}$ and $\{s_{k}\}_{k\geq 1}$ such that, for each $k\geq 1,$
We readily check that
Without loss of generality, we assume that $n_{k+1}-n_{k}>s_{k}$ for all $k\geq 1.$ Otherwise, we consider only sufficiently large k. Put
where
Define a marked set $\mathbb {K}$ of positive integers by
Now we define m sequences as follows.
-
• For $i=1,$
$$ \begin{align*}a_{n_{k}}^{(1)}=1,\ a_{n_{k}+1}^{(1)}=\cdots=a_{n_{k}+s_{k}-1}^{(1)}=1,\ a_{n_{k}+s_{k}}^{(1)}=a_{n_{k}+2s_{k}}^{(1)}= \cdots=a_{n_{k}+\iota_ks_{k}}^{(1)}=1.\end{align*} $$ -
• For $2\leq i\leq m,$
$$ \begin{align*}a_{n_{k}}^{(i)}=i,\ a_{n_{k}+1}^{(i)}=\cdots=a_{n_{k}+s_{k}-1}^{(i)}=1,\ a_{n_{k}+s_{k}}^{(i)}=a_{n_{k}+2s_{k}}^{(i)}= \cdots=a_{n_{k}+\iota_ks_{k}}^{(i)}=i.\end{align*} $$
Then, for $i=1,2,\ldots ,m,$ write
Now we prove $\prod _{i=1}^{m}E(\mathbb {K},\{a_{n}^{(i)}\}_{n\geq 1})\subset E(\alpha ,\beta ).$ Fix $(x_{1},\ldots ,x_{m})\in \prod _{i=1}^{m}E(\mathbb {K},\{a_{n}^{(i)}\}_{n\geq 1})$ for any $n\geq n_{1}$ and let k be the integer such that $n_{k}\leq n< n_{k+1}.$ From the construction of the set $\prod _{i=1}^{m}E(\mathbb {K},\{a_{n}^{(i)}\}_{n\geq 1})$ , we see that
Further, by (4.1), we deduce that
and
Hence, $(x_{1},\ldots ,x_{m})\in E(\alpha ,\beta ).$
It remains to prove that the density of $\mathbb {K}\subset \mathbb {N}$ is zero. For $n_{k}\leq n<n_{k+1}$ with some $k\geq 1{:}$
-
• if $n_{k}\leq n< n_{k}+s_{k},$ then $\sharp \{i\leq n : i\in \mathbb {K}\}=\sum _{j=1}^{k-1}(m_{j}+\iota _{j})+n-n_{k}+1;$
-
• if $n_{k}+ls_{k}\leq n<n_{k}+(l+1)s_{k}$ for some $l$ with $0< l<\iota _{k},$ then we see that ${\sharp \{i\leq n : i\in \mathbb {K}\}=\sum _{j=1}^{k-1} (s_{j}+\iota _{j})+s_{k}+l};$
-
• if $n_{k}+\iota _{k}s_{k}\leq n<n_{k+1},$ then $\sharp \{i\leq n : i\in \mathbb {K}\}=\sum _{j=1}^{k}(s_{j}+\iota _{j}).$
Consequently,
where the last equality follows by the Stolz–Cesàro theorem and (4.2). By Lemmas 2.4 and 2.5,
Similar arguments apply to the remaining cases. We only give the constructions for the proper sequences $\{n_k\}_{k\geq 1}$ and $\{s_k\}_{k\geq 1}.$
Case 2: $0<\alpha =\beta <\infty .$ Take $n_{k}=2^{k}$ and $s_{k}=\lfloor \alpha \log n_{k}\rfloor \text { for } k\geq 1.$
Case 3: $\alpha =0<\beta <\infty .$ Take $n_{k}=2^{2^{2^{k}}}$ and $s_{k}=\lfloor \,\beta \log n_{k}\rfloor \text { for } k\geq 1.$
Case 4: $\alpha =0, \beta =\infty .$ Take $n_{k}=2^{2^{2^{k}}}$ and $s_{k}=\lfloor k\log n_{k}\rfloor ~\text { for } k\geq 1.$
Case 5: $0<\alpha <\beta =\infty .$ Take $n_{k}=2^{k!}$ and $s_{k}=\lfloor \alpha k\log n_{k}\rfloor \text { for } k\geq 1.$
Case 6: $\alpha =\beta =0.$ Take $n_{k}=2^{k}$ and $s_{k}=\lfloor \log \log n_{k}\rfloor \text { for } k\geq 1.$
Case 7: $\alpha =\beta =\infty .$ Take $n_{k}=2^{k}$ and $s_{k}=\lfloor k\log n_{k}\rfloor \text { for } k\geq 1.$