Uniform Diophantine approximation and run-length function in continued fractions

We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\widehat{\nu },$ we calculate the Hausdorff dimension of the uniform Diophantine set $$\mathcal{U}(y,\widehat{\nu })=\{x\in [0,1)\text{\hspace{0.17em}⁣}:\mathrm{\forall }N\gg 1,\mathrm{\exists }n\in [1,N],\text{ such that }\mathrm{\mid }{T}^n(x)-y\mathrm{\mid }<\mathrm{\mid }{I}_N(y){\mathrm{\mid }}^{\widehat{\nu }}\}$$ for algebraic irrational points $y\in [0,1)$. These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions.