Hausdorff dimension of the Cartesian product of exact approximation set in $\beta$-expansions

In this paper, we study the metrical theory of Cartesian products of exact approximation sets in $\beta$-expansions. More precisely, for an integer $d\ge 2$ and real numbers ${\beta }_i>1$ $(1\le i\le d)$, we consider the set of points $x_i\in [0,1)$ is approximable by its convergents in the ${\beta }_i$-expansion to order ${\psi }_i$, but not to any better order. For any non-increasing functions ${\psi }_i$, we determine the Hausdorff dimension of the Cartesian product of these sets.