Higher dimensional shrinking target problem in beta dynamical systems

We consider the two dimensional shrinking target problem in the beta dynamical system for general $\beta>1$ and with the general error of approximations. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in[0,1]$, define the shrinking target set $$E(T_\beta, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{ll} |T_{\beta}^{n}x-x_{0}|<e^{-S_nf(x)}\\ [1ex] |T_{\beta}^{n}y-y_{0}|< e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \N \right\},$$ where $S_nf(x)=\sum_{j=0}^{n-1}f(T_\beta^jx)$ is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to some pressure function. This represents the first result of this kind for the higher dimensional beta dynamical systems.