The theory of continued fractions is an extremely useful tool in approximating irrational numbers by rational numbers. Any number $x\in \mathbb {R}{\setminus} \mathbb {Q}$ can be uniquely represented by a continued fraction of the form

$$ \begin{align*} x=a_0(x)+\cfrac{1}{a_1(x)+\cfrac{1}{a_2(x)+\cfrac{1}{\ddots}}}=[a_0(x); a_1(x),a_2(x),\ldots], \end{align*} $$

where $a_n(x)\in \mathbb {Z}, a_n(x)\geq 1$ for $n\geq 1$ , is known as the nth partial quotient of x. The classical theory of continued fractions shows that the convergents of the partial quotients of x give exactly the best rational approximation of x (see [Reference Khinchin11, Theorems 16 and 17]. The nth convergent is given by

$$ \begin{align*} \frac{p_n}{q_n}:=[a_0(x); a_1(x),\ldots, a_n(x)], \end{align*} $$

where $p_n,q_n\in \mathbb {Z}$ are coprime and $q_n\geq 1$ . The speed of approximation for any irrational number x is related to the size of the partial quotients by

$$ \begin{align*} \bigg| x - \frac{p_n}{q_n}\bigg| < \frac{1}{q_n(a_{n+1}q_n + q_{n-1})} \quad \text{for all } n\in\mathbb{N}. \end{align*} $$

Kleinbock and Wadleigh [Reference Kleinbock and Wadleigh12] showed that Dirichlet’s theorem is optimal in a precise sense. For any nonincreasing function $\psi :\mathbb {N}\to \mathbb {R}_+$ , define the set of $\psi $ -Dirichlet improvable numbers by

$$ \begin{align*} D(\psi):=\bigg\{x\in\mathbb R: \begin{array}{ll}\exists\, N \ {\rm such\ that\ the\ system}\ |qx-p|\, <\, \psi(t), |q|<t\ \\ \text{has a nontrivial integer solution for all }t>N\end{array}\!\!\bigg\}. \end{align*} $$

Then, Kleinbock and Wadleigh showed for $x\in [0,1) \setminus \mathbb {Q}$ that:

1. (i) $x\in D(\psi )$ if $a_{n+1}(x)a_n(x)\, \le \,\psi (q_n)/4$ for all sufficiently large n;

2. (ii) $x\not \in D(\psi )$ if $a_{n+1}(x)a_n(x)\,>\, \psi (q_n)$ for infinitely many n.

The metric theory for the set $D(\psi )$ is fully characterised in the papers [Reference Bos, Hussain and Simmons2, Reference Huang, Wu and Xu8, Reference Hussain, Kleinbock, Wadleigh and Wang9].

My thesis contains results on the metric theory of continued fraction and Lüroth series expansions. The first result gives metrical properties of the product of partial quotients in the plane. Let $\Psi :\mathbb N\to \mathbb R_+$ be a function. Define the set, for $(t_1, \ldots , t_m)\in \mathbb R_{+}^m$ ,

$$ \begin{align*} \Lambda(\Psi):=\bigg\{(x, y)\in[0,1]^2:\max\bigg\{\prod_{i=1}^ma_{n+i}^{t_i}(x), \prod_{i=1}^ma_{n+i}^{t_i}(y)\bigg\} \geq \Psi(n) \ \text{for all} \ n\geq 1\bigg\}. \end{align*} $$

For the one-dimensional analogue of this set, the Hausdorff dimension (for $m=2$ ) was determined in [Reference Bakhtawar, Hussain, Kleinbock and Wang1] and can also be deduced from [Reference Hussain and Shulga10]. In my thesis, I prove the following two-dimensional result. Throughout, $\dim _{H}$ is the Hausdorff dimension.

Theorem 1 [Reference Brown-Sarre and Hussain5].

Let $\Psi $ be a positive function. Then,

$$ \begin{align*} \dim_{H} (\Lambda(\Psi))=\frac{2+\tau}{1+\tau}\quad \text{where } \log\tau=\limsup_{n\to\infty}\frac{\log\log\Psi(n)}{n}.\end{align*} $$

For a nondecreasing function $\varphi : \mathbb {N} \to [2,\infty )$ and $\ell \in \mathbb {N}$ , define the set

$$ \begin{align*} \mathcal{F}_{\ell}(\varphi) := \bigg\{ x\in [0,1): \begin{matrix} a_{j}(x)\cdots a_{j+\ell-1}(x) \geq \varphi(n) \\ a_{k}(x)\cdots a_{k+\ell-1}(x) \geq \varphi(n) \end{matrix}\; \text{ with } 1\leq j < k \leq n \text{ for i.m. } n\in\mathbb{N}\bigg\}, \end{align*} $$

where ‘i.m.’ stands for ‘infinitely many’. The set $\mathcal {F}_{\ell }(\varphi )$ arises in the determination of laws of large numbers for partial quotients. Phillip [Reference Philipp13] proved that there is no reasonable function $\sigma :\mathbb {N}\to \mathbb {R}_+$ such that ${(a_1(x)+a_2(x)+\cdots +a_n(x))}/{\sigma (n)}$ converges almost everywhere as $n \to \infty $ . However, Diamond and Vaaler [Reference Diamond and Vaaler6] showed that such a relation holds if we omit the largest partial quotient. Hu et al. [Reference Hu, Hussain and Yu7] extended this further by proving the case for the sum of products of two consecutive partial quotients and omitting the largest product. They proved that almost every $x\in [0,1)$ satisfies

(1) $$ \begin{align} \lim_{n\to\infty} \cfrac{1}{n\log^2 n}\hspace{0.5pt}\bigg( \sum_{j=1}^n a_j(x)a_{j+1}(x) - \max_{1\leq j \leq n} a_j(x)a_{j+1}(x)\bigg) = \cfrac{1}{2\log 2}. \end{align} $$

This led Tan et al. in [Reference Tan, Tian and Wang14] and Tan and Zhou in [Reference Tan and Zhou15] to find a zero-one law for the Lebesgue measure of $\mathcal {F}_1(\varphi )$ . We extend this work to $\mathcal {F}_{3}(\varphi )$ .

Theorem 2 [Reference Brown-Sarre, González Robert and Hussain4].

Let $\varphi :\mathbb {N}\to [2,\infty )$ be nondecreasing. The Lebesgue measure $\lambda $ of $\mathcal {F}_{3}(\varphi )$ is given by

$$ \begin{align*} \lambda(\mathcal{F}_{3}(\varphi)) = \begin{cases} 0 &\text{if } \displaystyle\sum_{n\geq 1} \frac{n\log^{4}\varphi(n)}{\varphi^2(n)} + \frac{\log \varphi(n)}{\varphi(n)}<\infty, \\ 1 &\text{if } \displaystyle\sum_{n\geq 1} \frac{n\log^{4}\varphi(n)}{\varphi^2(n)} + \frac{\log \varphi(n)}{\varphi(n)}=\infty. \end{cases} \end{align*} $$

I further calculate the Hausdorff dimension for $\mathcal {F}_3(\varphi )$ . Define $g_3:\mathbb {R}\to \mathbb {R}$ by

$$ \begin{align*} g_3(s) := \frac{3s^3-5s^2+4s-1}{s^2-s+1}. \end{align*} $$

For a function $\varphi :\mathbb {N}\to \mathbb {R}_+$ , let B and b be defined by

(2) $$ \begin{align} \log B = \liminf_{n\to\infty} \frac{\log \varphi(n)}{n} \quad\text{and}\quad \log b = \liminf_{n\to\infty} \frac{\log\log \varphi(n)}{n}. \end{align} $$

Theorem 3 [Reference Brown-Sarre, González Robert and Hussain4].

Let $\varphi :\mathbb {N}\to [2,\infty )$ be nondecreasing. Then, the Hausdorff dimension of $\mathcal {F}_3(\varphi )$ is given by

$$ \begin{align*} \dim_{H} \mathcal{F}_3(\varphi) = \begin{cases} 1 &\text{if } B=1, \\ \inf\{s \geq 0: P(T,-g_3(s)\log B - s \log |T'|) \leq 0\} &\text{if } 1<B<\infty, \\ {1}/{(1+b)} &\text{if } B=\infty, \end{cases} \end{align*} $$

where $P(T,\cdot )$ is a pressure function.

The thesis also contains a result on the Lebesgue measure of a set associated with the Lüroth series expansion of a real number. Every $x\in (0,1]$ has a Lüroth series expansion

$$ \begin{align*} x= \frac{1}{d_1} + \frac{1}{d_1(d_1-1)d_2} + \frac{1}{d_1(d_1-1)d_2(d_2-1)d_3} + \cdots \end{align*} $$

with a unique sequence $(d_n)_{n\geq 1}$ of integers at least $2$ . Let $m\in \mathbb {N}$ , $\mathbf {t}=(t_0,\ldots , t_{m-1})\in \mathbb {R}_{+}^m$ and $\liminf _{n\to \infty }\Psi (n)>1$ . Define the set

$$ \begin{align*} \mathfrak{E}_{\mathbf{t}}(\Psi) := \bigg\{ x\in [0,1): \prod_{i=0}^{m-1} d_{n+i}^{t_i}(x) \geq \Psi(n) \text{ for infinitely many }n\in\mathbb{N}\bigg\}, \end{align*} $$

and the numbers

$$ \begin{align*} t_{\min} := \min\{t_0,t_1,\ldots, t_{m-1}\}, \quad \; t_{\max} :=\max\{t_0,t_1,\ldots, t_{m-1}\} \end{align*} $$

and

$$ \begin{align*} \ell(\mathbf{t}):=\# \{j\in \{0,\ldots, m-1\}: t_j=t_{\max}\}. \end{align*} $$

Theorem 4 [Reference Brown-Sarre, González Robert and Hussain3].

Let $m\in \mathbb {N}$ and $\mathbf {t}\in \mathbb {R}_{+}^m$ be arbitrary. If $\liminf _{n\to \infty } \Psi (n)>1,$ then

(3) $$ \begin{align} \lambda\left(\mathfrak{E}_{\mathbf{t}}(\Psi)\right) = \begin{cases} 0 &\text{if } \displaystyle\sum_{n=1}^{\infty} \cfrac{\left( \log\Psi(n)\right)^{\ell(\mathbf{t}) - 1} }{\Psi(n)^{{1}/{t_{\max}}}} < \infty, \\[2ex] 1 &\text{if } \displaystyle\sum_{n=1}^{\infty} \cfrac{\left( \log\Psi(n)\right)^{\ell(\mathbf{t}) - 1}}{\Psi(n)^{{1}/{t_{\max}}}} = \infty. \end{cases} \end{align} $$

Theorem 5 [Reference Brown-Sarre, González Robert and Hussain3].

Let B and b be given by (2). For any $m\in \mathbb {N}$ and $\mathbf {t}\in \mathbb {R}_{+}^{m}$ ,

$$ \begin{align*} \dim_{H} \mathfrak{E}_{\mathbf{t}}(\Psi) = \begin{cases} 1 &\text{if } B=1, \\ {1}/{(b+1)} &\text{if } B=\infty. \end{cases} \end{align*} $$

Theorem 6 [Reference Brown-Sarre, González Robert and Hussain3].

Suppose $m=2$ . Let B and b be given by (2) and assume $1<B<\infty $ . For a given $\mathbf {t}=(t_0,t_1)\in \mathbb {R}_{+}^2$ , define

$$ \begin{align*} f_{t_0,t_1}(s):= \frac{s^2}{t_0t_1\max \{ {s}/{t_1} + {(1-s)}/{t_0}, {s}/{t_0}\}}. \end{align*} $$

Then, the Hausdorff dimension of $\mathfrak {E}_{\mathbf {t}}(\Psi )$ is the unique solution of

$$ \begin{align*} \sum_{d=2}^{\infty} \frac{1}{d^s(d-1)^s B^{f_{t_0,t_1}(s)}}=1. \end{align*} $$