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\title[Localized growth speed]{Some lacunarity properties of partial quotients of real numbers}

\alttitle{Quelques propriétés de lacunarité des quotients partiels de nombres réels}


\author{\firstname{Xuan} \lastname{Zhao}} 


\address{National Education Examinations Authority, 100084 Beijing, China}

\email{zhaox@mail.neea.edu.cn}


\author{\firstname{Zhenliang} \lastname{Zhang}\IsCorresp}

\address{School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, P. R. China}

\email{zhliang\_zhang@163.com}

\thanks{Zhenliang Zhang is supported by project of the Science and Technology Research Program of Chongqing Municipal Education Commission (No.~KJQN202100528), Natural Science Foundation of Chongqing (No.~CSTB2022NSCQ-MSX1255).}

\CDRGrant[Science and Technology Research Program of Chongqing Municipal Education Commission]{KJQN202100528}

\CDRGrant[Natural Science Foundation of Chongqing]{CSTB2022NSCQ-MSX1255}

\subjclass[2010]{11K50, 28A78}

\keywords {Hausdorff dimension, Continued fraction expansion.}

\altkeywords {Dimension de Hausdorff, développement en fraction continue.}


\begin{abstract}
We consider lacunarity properties of sequence of partial quotients for real numbers in their continued fraction expansions.
Hausdorff dimension of the sets of points with different lacunarity conditions on their partial quotients are calculated.
\end{abstract}

\begin{altabstract}
Nous considérons des propriétés de lacunarité de la suite des quotients partiels du développement en fraction continue de nombres réels. Nous calculons la dimension de Hausdorff d'ensembles de points dont la suite des quotients partiels satisfait à différentes conditions de lacunarité.
\end{altabstract}




\dateposted{2024-11-05}
\begin{document}


\maketitle

\section{Introduction}

Continued fraction expansion is induced by the Gauss transformation $T:[0,1)\to [0,1)$ given by 
\[
T(0)\coloneqq 0, \ \ T(x)=\frac{1}{x}{\text{mod}\ 1}, \ x\in (0,1).
\]
Then every irrational number $x\in[0,1)$ can be uniquely expanded into an infinite continued fraction:\mots{
\begin{align*}
x=\cfrac{1}{a_{1}(x)+\cfrac{1}{a_{2}(x)+\ddots}}\coloneqq [a_1(x),a_2(x),\ldots,],
\end{align*}
}%
where $a_{1}(x)=\lfloor 1/x\rfloor$ and $a_{n}(x)=a_{1}\left(T^{n-1}(x)\right)$ for $n\ge 2$ are called the partial quotients of $x$. The finite truncation \begin{align*}
\frac{p_{n}(x)}{q_{n}(x)}=\left[a_{1}(x),\ldots,a_{n}(x)\right]
\end{align*} is called the $n$th convergent of $x$.

It is \mots{well-known} that continued fraction expansion plays an important role in Diophantine approximation and dynamical systems:
\begin{itemize}
  \item In Diophantine approximation, 
  how well an irrational number can be approximated by rationals depends on the growth rate of the partial quotients. 
  For example, 
  the classic Jarn\'{i}k set \cite{Jar} can be expressed as
      \[
  \mathcal{K}(\psi)=\Big\{x\in [0,1): a_{n+1}(x)\ge \psi(q_n(x)), \ {\text{for infinitely many $n\in \N$}}\Big\}.
  \]

  \item In dynamical system, continued fraction is a classic dynamics with infinitely many branches \cite{MU99}.
\end{itemize}

As said above, 
the growth rate of the partial quotient is tightly related to the Diophantine properties of an irrational number. 
Many metric results have been achieved in this aspect, such as the Borel--Bernstein theorem \mots{that} 
deals with \mots{Lebesgue} measure theory on the growth rate of 
\mots{$\{a_n(x)\}_{n\ge 1}$. 
Hausdorff dimension
} 
of sets obeying some restrictions on the partial quotients have been well established in Good \cite{Good}, {\L}cuzak \cite{Luczak}, Wang \& Wu \cite{Wangwu}, etc.

\mots{
Very recently}, it was found by Kleinbock and Wadleigh \cite{KlWa} 
that the Dirichlet improvable set is highly related to the growth rate of the product of two consecutive partial quotients in the following sense.
\mots{
Let $\psi\colon [t_0,\infty) \to \mathbb{R}_+$ be non-increasing with $t_0>1$ is fixed and $t\psi(t)<1$ for all $t\geq t_0$.}
The $\psi$-Dirichlet improvavable set is defined as
\[
\mathcal{D}(\psi)=\Big\{x\in [0,1): \min_{1\le q<Q}\|qx\|<\psi(Q), \ {\text{for all $Q\gg 1$}}\Big\}.
\]
Then by taking $\Psi(q)=\frac{q\psi(q)}{1-q\psi(q)}$,
one has
\[
G(\Psi)\subset [0,1)\setminus \mathcal{D}(\psi)\subset G\left(\frac{\Psi}{4}\right)
\]
where
\[
G(\Psi)=\Big\{x\in [0,1): a_{n+1}(x)a_n(x)\ge \Psi(q_n(x)), \ {\text{for infinitely many}}\ n\in \N\Big\}.
\]

Since \mots{then,} 
the metric theory relating the growth of two consecutive partial quotients are extensively studied in \mots{
\cite{Ba1,Ba2,PhilipMumtazDavid,Huang,HKWW,MumtazLiShulga,MumtazShulga,LWX}.}

In this note, we take another turn by studying the relative growth rate of two consecutive partial quotients.
All the above mentioned works are concerning the size of limsup sets,
however,
the sets we will consider below are of liminf nature.
So the method used here is different from the above works (see \cite{MumtazShulga} 
for a unified way \mots{of} dealing with the Hausdorff dimension of the above mentioned works).
These liminf sets concerns the points with partial quotients increasing very fast,
so can be uniformly well approximated by their convergents.
At first, we give some notations.
Let $\{s_n\}_{n\ge 1}$ be a sequence of strictly increasing sequence of integers.
\begin{itemize}
\item call it a sub-lacunary sequence, if
\[ 
\lim_{n\to\infty}s_{n+1}/s_n=1;
\]

\item call it lacunary if $\exists c>1$ such that
\[
s_{n+1}/s_n\ge c, \ {\text{for all}}\ n\gg 1;
\]
\begin{itemize}
\item log-lacunary if for some $t>0$,
\[
\frac{s_{n+1}}{s_n}\ge (\log n)^t, \ {\text{for all}}\ n\gg 1;\ \ \ \
\]
\item polynomial-lacunary if for some $t>0$,
\[
\frac{s_{n+1}}{s_n}\ge n^t, \ \     \ {\text{for all}}\ n\gg 1;
\]
\item exponential-lacunary if  for some $b>1$,
\[
\frac{s_{n+1}}{s_n}\ge b^n, \ \    \mots{\ {\text{for all}}\ n\gg 1.}
\]
  \end{itemize}
\end{itemize}
At first,
we give an auxiliary set to be compliant with the set of points with strictly increasing partial quotients.
\begin{equation}\label{f1}
G=\Big\{x\in [0, 1): \{a_n(x)\} \ {\text{is a strictly increasing sequence}}\Big\}.
\end{equation}
Then we define \begin{align*}
  {\mathcal{S}}_G&=\Big\{x\in G: \{a_n(x)\}\  {\text{is sub-lacunary}}\Big\},\\
  {\mathcal{L}}_G&=\Big\{x\in G: \{a_n(x)\}\  {\text{is log-lacunary}}\Big\},\\
  {\mathcal{P}}_G&=\Big\{x\in G: \{a_n(x)\}\  {\text{is polynomial-lacunary}}\Big\},\\
  {\mathcal{E}}_G&=\Big\{x\in G: \{a_n(x)\}\  {\text{is exponential-lacunary}}\Big\}.
\end{align*}
In this note, we show that 

\begin{theo}\label{t1} By denoting $\hdim$ the Hausdorff dimension, \mots{we have}
  \[
  \hdim {\mathcal{S}}_G=\hdim {\mathcal{L}}_G=\hdim {\mathcal{P}}_G=\hdim {\mathcal{E}}_G=1/2.
  \]
\end{theo}










\section{Preliminaries}\label{sec2}


Recall that $p_n(x)/q_n(x)$ is the $n$th convergent of $x$. The numerator and denominator of $p_n(x)/q_n(x)$ can be determined recursively:
for any $k\ge 1$
\begin{equation}\label{rule}
\begin{split}
p_{k}(x)=a_{k}(x)p_{k-1}(x)+p_{k-2}(x),\ \
q_{k}(x)=a_{k}(x)q_{k-1}(x)+q_{k-2}(x)
\end{split}
\end{equation}
with the conventions $p_{0}=0$, $q_{0}=1$, $p_{-1}=1$, $q_{-1}=0$.

For simplicity, we write
\begin{equation}\label{ff6}p_n(x)=p_{n}\left(a_{1},\ldots,a_{n}\right)=p_{n}, \ \ q_{n}(x)=q_n\left(a_{1},\ldots,a_{n}\right)=q_{n}\end{equation}
when the partial quotients $a_1,\ldots, a_n$ are clear. %In the following, without additional announcement,


For any positive integers $a_{1},\ldots,a_{n}$, define
\begin{align*}
I_{n}\left(a_{1},\ldots,a_{n}\right)\coloneqq \left\{x\in\left[0,1\right):a_{1}\left(x\right)=a_{1},\ldots,a_{n}\left(x\right)=a_{n}\right\}
\end{align*} and call it $\textit{a cylinder of order }n$. We use $I_n(x)$ to denote the $n$th order cylinder containing $x$.

\begin{prop}[Khinchin~\cite{Khin2}]\label{range}
For any $n\ge 1$ and $\left(a_{1},\ldots,a_{n}\right)\in\mathbb{N}^{n}$, $p_{k},q_{k}$ are defined recursively by $\left(\ref{rule}\right)$ for $0\le k\le n$. Then
\begin{equation}I_{n}\left(a_{1},\ldots,a_{n}\right)=
  \begin{cases}
   \left[\dfrac{p_{n}}{q_{n}},\dfrac{p_{n}+p_{n-1}}{q_{n}+q_{n-1}}\right) &\mots{{\rm if }}\,\,n\textrm{ is even}\\
   \left(\dfrac{p_{n}+p_{n-1}}{q_{n}+q_{n-1}},\dfrac{p_{n}}{q_{n}}\right] &\mots{{\rm if }}\,\,n\textrm{ is odd}.\\
  \end{cases}
\end{equation}
Therefore, the length of a cylinder of order $n$ is given by
\begin{equation*}\label{length}
\left|I_{n}\left(a_{1},\ldots,a_{n}\right)\right|=\dfrac{1}{q_{n}\left(q_{n}+q_{n-1}\right)}.
\end{equation*}
\end{prop}

The next lemma relates a ball with the cylinders, basically following from Proposition \ref{range} on the distribution of cylinders \cite{IosK}.
\begin{prop}\label{l2.5}
  Let $x\in I_n(a_1,\ldots,a_n)$ with $a_n\ge 2$. Then 
\[
  B(x, |I_n(a_1,\ldots,a_n)|)\subset \bigcup_{i=-1}^2I_n(a_1,\ldots,a_n+i).
  \]
\end{prop}

Next, we introduce the mass distribution principle which is a classic method in estimating the Hausdorff dimension of a set from below.
\begin{prop}[\cite{Fal}]
\label{MD}
 Let $E$ be a Borel set and $\mu$ be a measure with $\mu\left(E\right)>0$. Suppose that for any $x\in E$,
\begin{equation}\label{MA}
\liminf_{r\to 0}\frac{\log \mu(B(x,r))}{\log r}\ge s
\end{equation}
where $B\left(x,r\right)$ denotes an open ball centered at $x$ and radius $r$, then $\hdim E\ge s$.
\end{prop}






\section{Proof of Theorem \ref{t1}}

Good \cite{Good} showed that the Hausdorff dimension of $G$ defined in (\ref{f1}) \mots{is one-half}, 
so by the simple inclusion 
\[
G\supset {\mathcal{S}}_G,\ {\text{and}}\ \ G\supset {\mathcal{L}}_G\supset {\mathcal{P}}_G\supset {\mathcal{E}}_G,
\]
 it is sufficient to show that 
\[
\hdim {\mathcal{S}}_G\ge 1/2 \ {\text{and}}\ \hdim {\mathcal{E}}_G\ge 1/2.
\]

\begin{lemma}
  Let $\alpha>1$. Define 
\[
  E_{\alpha}=\Big\{x\in [0,1): (2n-1)^{\alpha}\le a_n(x)<(2n)^{\alpha}, \ {\text{for all}}\ n\ge 1\Big\}.
  \]
 Then 
\[
  \hdim E_{\alpha}=\frac{\alpha-1}{2\alpha}.
  \]
\end{lemma}
\begin{proof}\ 

\begin{enumerate}[(\Roman*)]
\item  For the upper bound of $\hdim E_{\alpha}$, we consider a natural cover of $E_{\alpha}$. It is clear that for any $N\ge 1$, the family of intervals 
\[
  \bigcup_{a_1,\ldots, a_N: (2n-1)^{\alpha}\le a_n(x)<(2n)^{\alpha},  \ 1\le n\le N} I_N(a_1,\ldots, a_N)
  \]
 covers $E_{\alpha}$. Therefore, for any $s>0$, the $s$-Hausdorff measure of $E_{\alpha}$ can be estimated from above~by
  \begin{align*}
    \mathcal{H}^s(E_{\alpha})&\le \liminf_{N\to\infty}\sum_{a_1,\ldots, a_N: (2n-1)^{\alpha}\le a_n(x)<(2n)^{\alpha}, 1\le n\le N}|I_N(a_1,\ldots,a_n)|^s\\
    &\le \liminf_{N\to\infty}\sum_{a_1,\ldots, a_N: (2n-1)^{\alpha}\le a_n<(2n)^{\alpha}, 1\le n\le N}\prod_{n=1}^N\frac{1}{a_n^{2s}}\\
    &=\liminf_{N\to \infty}\prod_{n=1}^N\sum_{(2n-1)^{\alpha}\le a_n<(2n)^{\alpha}}\frac{1}{a_n^{2s}}.
  \end{align*}
  For any $s>\frac{\alpha-1}{2\alpha}$, one can choose $n_o$ such that for all $n\ge n_o$, 
\[
  \sum_{(2n-1)^{\alpha}\le a_n<(2n)^{\alpha}}\frac{1}{a_n^{2s}}\le \frac{(2n)^{\alpha}-(2n-1)^{\alpha}}{(2n-1)^{2\alpha s}}\le \frac{2\alpha \cdot (2n)^{\alpha-1}}{(2n-1)^{2\alpha s}}<1,
  \]
 and  thus it follows that 
\[
  \mathcal{H}^s(E_{\alpha})\le \prod_{n=1}^{n_o}\sum_{(2n-1)^{\alpha}\le a_n<(2n)^{\alpha}}\frac{1}{a_n^{2s}}<\infty.
  \]
 As a result, 
\[
\hdim E_{\alpha}\le \frac{\alpha-1}{2\alpha}.
\]

\item 
For the lower bound of $\hdim E_{\alpha}$, notice that the set $E_{\alpha}$ has a nice Cantor structure. For each $n\ge 1$, let 
\[
\mathcal{E}_n=\Big\{I_n(a_1,\ldots, a_n): (2k-1)^{\alpha}\le a_k<(2k)^{\alpha},\ \ {\text{for all}}\ 1\le k\le n\Big\}.
\]
  Then 
\[
E=\bigcap_{n=1}^{\infty}\bigcup_{I_n(a_1,\ldots, a_n)\in \mathcal{E}_n}I_n(a_1,\ldots, a_n),
\]
 and every element in $I_{n-1}(a_1,\ldots, a_{n-1})$ in $\mathcal{E}_{n-1}$ contains exactly 
\[
c_1 n^{\alpha-1}\le D_{n}\coloneqq (2n)^{\alpha}-(2n-1)^{\alpha}\le c_2 n^{\alpha-1}.
\]
 elements $I_n(a_1,\ldots, a_n)$ in $\mathcal{E}_{n}$.

\mots{Then, we distribute a mass supported} on $E_{\alpha}$ by setting 
\[
\mu(I_n(a_1,\ldots,a_n))=\frac{1}{D_1\cdots D_n},
\]
 for all $n\ge 1$ and $I_n(a_1,\ldots, a_n)$ in $\mathcal{E}_{n}$.

For any $x\in E_{\alpha}$, 
for each $r>0$ small enough, let $n$ be the integer such that 
\[
|I_{n+1}(x)|<r\le |I_n(x)|.
\] 
\mots{Then, by Proposition \ref{l2.5},} the ball $B(x,r)$ can intersect at most three cylinders of order $n$, thus \begin{align*}
  \liminf_{r\to\infty}\frac{\log \mu(B(x,r))}{\log r}\ge \liminf_{n\to \infty}\frac{\log [3\cdot (D_1\cdots D_n)^{-1}]}{\log |I_{n+1}(x)|}.
\end{align*}
By the recursive relation of $q_n(x)$, it follows that 
\[
q_{n+1}(x)\le \prod_{k=1}^{n+1}(a_k(x)+1)\le \prod_{k=1}^{n+1}(2k)^{\alpha}\le c^n \prod_{k=1}^{n+1}k^{\alpha};
\]
 on the other hand, 
\[
\prod_{k=1}^nD_k\ge c_1^n\prod_{k=1}^n k^{\alpha-1}.
\]
 Thus it follows that 
\[
\liminf_{r\to 0}\frac{\log \mu((x,r))}{\log r}\ge \liminf_{n\to \infty}\frac{\log \prod_{k=1}^n k^{\alpha-1}}{\log \left(\prod_{k=1}^{n+1}k^{\alpha}\right)^2}=\frac{\alpha-1}{2\alpha}.
\]
 Finally by Proposition \ref{MD}, one has 
\[
\hdim E_{\alpha}\ge \frac{\alpha-1}{2\alpha}.\qedhere
\]
\end{enumerate}
\end{proof}

By a result in \cite{FLWW} or the same line of the argument as above, except minor modifications on notation, one can show the following. \begin{lemma}
  Let $b>1$. Define 
\[
  F_{b}=\Big\{x\in [0,1): b^{1+2+\cdots +n}\le a_n(x)<2\cdot b^{1+2+\cdots +n}, \ {\text{for all}}\ n\ge 1\Big\}.
  \]
 Then $
  \hdim F_b={1}/{2}.
  $
\end{lemma}



\begin{proof}[Proof of Theorem~\ref{t1}]  
It is clear that for any $\alpha>1$, 
\[
E_{\alpha}\subset \mathcal{S}_G,\ {\text{and so}}, \ \hdim \mathcal{S}_G\ge \lim_{\alpha\to \infty}\frac{\alpha-1}{2\alpha}=\frac{1}{2}.
\]
 It is also clear that for any $b>1$,
\[
F_{b}\subset \mathcal{E}_G,\ {\text{and so}}, \ \hdim \mathcal{E}_G\ge \frac{1}{2}.\qedhere
\]
\end{proof}


\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds from
any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

\printbibliography


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