We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, we calculate the Hausdorff dimension of the uniform Diophantine set

$$ \begin{align*} {\mathcal{U}(\hat{\nu})}= &\ \{x\in[0,1)\colon \text{for all }N\gg1,\text{ there exists }n\in[1,N],\\&\ \ \text{ such that }|T^{n}(x)-y| < |I_{N}(y)|^{\hat{\nu}}\} \end{align*} $$

$y\in [0,1)$

Let $ T:[0,1]\to [0,1] $ be an expanding Markov map with a finite partition. Let $ \mu _\phi $ be the invariant Gibbs measure associated with a Hölder continuous potential $ \phi $. For $ x\in [0,1] $ and $ \kappa>0 $, we investigate the size of the uniform approximation set

$$ \begin{align*}\mathcal U^\kappa(x):=\{y\in[0,1]:\text{ for all } N\gg1, \text{ there exists } n\le N, \text{ such that }|T^nx-y|<N^{-\kappa}\}.\end{align*} $$

The critical value of $ \kappa $ such that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)=1 $ for $ \mu _\phi $-almost every (a.e.) $ x $ is proven to be $ 1/\alpha _{\max } $, where $ \alpha _{\max }=-\int \phi \,d\mu _{\max }/\int \log |T'|\,d\mu _{\max } $ and $ \mu _{\max } $ is the Gibbs measure associated with the potential $ -\log |T'| $. Moreover, when $ \kappa>1/\alpha _{\max } $, we show that for $ \mu _\phi $-a.e. $ x $, the Hausdorff dimension of $ \mathcal U^\kappa (x) $ agrees with the multifractal spectrum of $ \mu _\phi $.

Let $r=[a_1(r), a_2(r),\ldots ]$ be the continued fraction expansion of a real number $r\in \mathbb R$. The growth properties of the products of consecutive partial quotients are tied up with the set admitting improvements to Dirichlet’s theorem. Let $(t_1, \ldots , t_m)\in \mathbb R_+^m$, and let $\Psi :\mathbb {N}\rightarrow (1,\infty )$ be a function such that $\Psi (n)\to \infty $ as $n\to \infty $. We calculate the Hausdorff dimension of the set of all $ (x, y)\in [0,1)^2$ such that

$$ \begin{align*} \max\left\{\prod_{i=1}^ma_{n+i}^{t_i}(x), \prod_{i=1}^ma_{n+i}^{t_i}(y)\right\} \geq \Psi(n) \end{align*} $$

$n\geq 1$

Let $\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$th partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$th convergent. The set of $\unicode[STIX]{x1D6F9}$-Dirichlet non-improvable numbers,

$$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$

$1/q^{2}\unicode[STIX]{x1D6F9}(q)$

${\mathcal{K}}(\unicode[STIX]{x1D6F9})$

${\mathcal{K}}(3\unicode[STIX]{x1D6F9})\subset G(\unicode[STIX]{x1D6F9})$

$s$

$s\in (0,1)$

$G(\unicode[STIX]{x1D6F9})\setminus {\mathcal{K}}(3\unicode[STIX]{x1D6F9})$

${\mathcal{K}}(\unicode[STIX]{x1D6F9})$

$G(\unicode[STIX]{x1D6F9})$