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, we calculate the Hausdorff dimension of the uniform Diophantine set align* U(ν̂)= &\ ∈[0,1) for all N1, there exists n∈[1,N],&\ \ such that |Tⁿ(x)-y| < |IN(y)|ν̂\ align* for a class of quadratic irrational numbers y∈ [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.