Approximation orders of real numbers in beta-dynamical systems

For any real numbers $x\in [0,1]$ and $\beta >1$, denote by $S_n(x,\beta )$ the partial sum of the first n terms in the β-expansion of x. It is known that for any $\beta >1$ and almost all $x\in [0,1]$, or for any $x\in (0,1]$ and almost all $\beta >1$, the approximation order of x by $S_n(x,\beta )$ is ${\beta }^{-n}$. Let $\phi :N\to R^{+}$ be a positive function. In this paper, we study the Hausdorff dimensions of the following two sets$A_{\beta }\left(\phi \right)=\{x\in [0,1]:\underset{n\to \infty }{\lim \sup }\frac{{\log }_{\beta }(x-S_n(x,\beta \left)\right)}{\phi \left(n\right)}=-1\},A_x\left(\phi \right)=\{\beta >1:\underset{n\to \infty }{\lim \sup }\frac{{\log }_{\beta }(x-S_n(x,\beta \left)\right)}{\phi \left(n\right)}=-1\},$ and complement the dimension theoretic results of these sets in [3], [6] and [18].