Increasing rate of weighted product of partial quotients in continued fractions

Each irrational number $x\in[0,1)$ admits a unique continued fraction expansion of the from

where for each $n\geq 1$, the positive integers $a_{n}(x)$ are known as the partial quotients of $x.$ The partial quotients can be generated by using the Gauss transformation $T:[0,1)\rightarrow[0,1)$ defined as

In fact, let $a_{1}(x)=\big{\lfloor}\frac{1}{x}\big{\rfloor}$ (where $\lfloor\cdot\rfloor$ denotes the integral part of real number). Then $a_{n}(x)=\big{\lfloor}\frac{1}{T^{n-1}(x)}\big{\rfloor}$ for $n\geq 2$. Sometimes (1.1) is written as $x=[a_{1}(x),a_{2}(x),a_{3}(x),\ldots].$ Further, the $n$-th convergent $p_{n}(x)/q_{n}(x)$ of $x$ is defined by $p_{n}(x)/q_{n}(x)=[a_{1}(x),a_{2}(x),\ldots,a_{n}(x)].$

The metrical aspect of the theory of continued fractions has been very well studied due to its close connections with Diophantine approximation. For example, for any $\tau>0$ the famous Jarník-Besicovitch set

can be described by using continued fractions. In fact,

For further details about this connection we refer to [11]. Thus the growth rate of the partial quotients reveals how well a real number can be approximated by rationals.

The well-known Borel-Bernstein Theorem [3, 4] states that for Lebesgue almost all $x\in[0,1),$ $a_{n}(x)\geq\varphi(n)$ holds for finitely many $n^{\prime}s$ or infinitely many $n^{\prime}s$ according to the convergence or divergence of the series $\sum_{n=1}^{\infty}{1}/{\varphi(n)}$ respectively. However, for rapidly growing function ${\varphi},$ the Borel-Bernstein Theorem does not give any conclusive information other than Lebesgue measure zero. To distinguish the sizes of zero Lebesgue measure sets, Hausdorff dimension is considered as an appropriate conception and has gained much importance in the metrical theory of continued fractions. Jarník [14] proved that the set of real numbers with bounded partials quotients has full Hausdorff dimension. Later on, Good [11] showed that the Hausdorff dimension of the set of numbers whose partial quotients tend to infinity is one half.

After that, a lot of work has been done in the direction of improving Borel-Bernstein Theorem, for example, the Hausdorff dimension of sets when partial quotients $a_{n}(x)$ obeys different conditions has been obtained in [6, 7, 8, 17, 18, 19, 20].

Motivation for studying the growth rate of the products of consecutive partial quotients aroses from the works of Davenport-Schmidt [5] and Kleinbock-Wadleigh [16] where they considered improvements to Dirichlet’s theorem. Let $\psi:[t_{0},\infty)\rightarrow\mathbb{R}_{+}$ be a monotonically decreasing function, where $t_{0}\geq 1$ is fixed. Denote by $D(\psi)$ the set of all real numbers $x$ for which the system

has a nontrivial integer solution for all large enough $t$. A real number $x\in D(\psi)$ (resp. $x\in D(\psi)^{c}$) will be referred to as $\psi$-Dirichlet improvable (resp. $\psi$-Dirichlet non-improvable) number.

The starting point for the work of Davenport-Schmidt [5] and Kleinbock-Wadleigh [16, Lemma 2.2] is an observation that Dirichlet improvability is equivalent to a condition on the growth rate of the products of two consecutive partial quotients. Precisely, they observed that

Then

where i.m. stands for infinitely many.

In other words, a real number $x\in[0,1)\setminus\mathbb{Q}$ is $\psi$-Dirichlet improvable if and only if the products of consecutive partial quotients of $x$ do not grow quickly. We refer the reader to [1, 2, 10, 12, 13] for more metrical results related with the set of Dirichlet non-improvable numbers.

As a consequence of Borel-Bernstein Theorem, for almost all $x\in[0,1)$ there exists a subsequence of partial quotients tending to infinity with a linear speed. In other words, for Lebesgue almost every $x\in[0,1)$

Taking inspirations from the study of the growth rate of the products of consecutive partial quotients for the real numbers, in this paper we consider the growth rate of the products of the consecutive weighted partial quotients. More precisely, by [2, Theorem 1.4], we have for Lebesgue almost all $x\in[0,1)$

where $t_{\max}=\max\{t_{i}:0\leq i\leq m\}$. This paper is concerned with Hausdorff dimension of some exceptional sets of (1.4). Let $\varphi:\mathbb{N}\to\mathbb{R}_{+}$ be a function satisfying $\varphi(n)/n\to\infty$ as $n\to\infty$ and let $t_{i}\in\mathbb{R}_{+}\ (0\leq i\leq m)$. Define the sets

and

The study of the level sets about the growth rate of $\{a_{n}(x)a_{n+1}(x):n\geq 1\}$ relative to that of $\{q_{n}(x):n\geq 1\}$ was discussed in [12]. Let $m\geq 2$ be an integer and $\Psi:\mathbb{N}\to\mathbb{R}_{+}$ be a positive function. The Lebesgue measure and the Hausdorff dimension of the set

have been comprehensively determined by Huang-Wu-Xu [13]. Very recently the results of [13] were generalized by Bakhtawar-Hussain-Kleinbock-Wang [2] to a weighted generalization of the set (1.5). For more details we refer the reader to [2, 13].

Our main results are as follows.

In this section, we fix some notations and recall some known results in theory of continued fraction expansions.

For an irrational number $x\in[0,1)$, recall $a_{n}(x)$ is the $n$-th partial quotient of $x$ in its continued fraction expansion. The sequences $\{p_{n}(x)\}_{n\geq 1},$ $\{q_{n}(x)\}_{n\geq 1}$ are the numerator and denominator of the $n$-th convergent of $x$. It is well-known that $\{p_{n}(x)\}_{n\geq 1}$ and $\{q_{n}(x)\}_{n\geq 1}$ can be obtain by the following recursive relations (see [15]):

with the conventions $p_{-1}=1,\ q_{-1}=0,\ p_{0}=0$ and $q_{0}=1$.

For any $n$-tuple $(a_{1},\cdots,a_{n})\in\mathbb{N}^{n}$ with $n\geq 1$, we call

a cylinder of order $n$.

Note that $p_{n}(x)$ and $q_{n}(x)$ are determined by the first $n$ partial quotients of $x$. So all points in $I_{n}(a_{1},\cdots,a_{n})$ determine the same $p_{n}(x)$ and $q_{n}(x)$. Hence for simplicity, if there is no confusion, we write $a_{n}$, $p_{n}$ and $q_{n}$ to denote $a_{n}(x)$, $p_{n}(x)$ and $q_{n}(x)$ for $x\in I_{n}(a_{1},\cdots,a_{n})$ respectively.

The following lemma is a collection of basic facts on continued fractions which can be found in the book of Khintchine [15].

The next theorem, known as Legendre’s Theorem, connects 1-dimensional Diophantine approximation with continued fractions.

According to Legendre’s theorem if an irrational $x$ is “well” approximated by a rational $p/q$, then this rational must be a convergent of $x$. So, the continued fraction expansions is a quick and efficient tool for finding good rational approximations to real numbers. For more basic properties of continued fraction expansions, one can refer to [15]. We also give some auxiliary results on the Hausdorff dimension theory of continued fractions that will be used later.

Applying Lemma 2.3, we obtain the follwing corollary which will be useful for the upper bound estimation on $\dim_{\mathrm{H}}\underline{E}(\{t_{i}\}_{i=0}^{m},\varphi).$

Let $\varphi:\mathbb{N}\to\mathbb{R}_{+}$ be a positive function with $\varphi(n)\rightarrow\infty$ as $n\rightarrow\infty$. For any $(t_{0},\cdots,t_{m})\in\mathbb{R}^{m+1}_{+}$ with $t_{i}\geq 0$ and at least one $t_{i}\neq 0\ (0\leq i\leq m)$, we introduce the sets

and

In order to prove Theorem 1.1, we first give a complete characterization on the size of the sets $ND(\varphi)$ and ${ND}^{\prime}(\varphi)$ in terms of Hausdorff dimension.

We remark that recently Zhang ([21]) obtained the Hausdorff dimension results of $ND(\varphi)$ and ${ND}^{\prime}(\varphi)$ for the special case $t_{0}=t_{1}=1$, $t_{i}=0\ (i\geq 2).$