Uniform Diophantine approximation and run-length function in continued fractions

Abstract

We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\hat{\hat{I}}½,$ we calculate the Hausdorff dimension of the uniform Diophantine set $$\mathcal{U}(y,\hat{\hat{I}}½)=\{x\in [0,1):\forall N\gg 1,\exists n\in [1,N],\text{ such that }|T^n(x)-y|<|I_N(y){|}^{\hat{\hat{I}}½}\}$$ 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.

33 pages, any comments for improvements are appreciated