Approximation orders of real numbers by β-expansions

We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their $\beta$-expansions with the exponential order $\beta^{-n}$. Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. These results are also applied to investigate the orbits of real numbers under $\beta$-transformation, the shrinking target type problem, the Diophantine approximation and the run-length function of $\beta$-expansions.