Uniform Diophantine approximation on the plane for $\beta$-dynamical systems

In this paper, we investigate the two-dimensional uniform Diophantine approximation in $\beta$-dynamical systems. Let ${\beta }_i>1(i=1,2)$ be real numbers, and let $T_{{\beta }_i}$ denote the ${\beta }_i$-transformation defined on $[0,1]$. For each $(x,y)\in [0,1{]}^2$, we define the asymptotic approximation exponent $$v_{{\beta }_1,{\beta }_2}(x,y)=\sup \{0\le v<\mathrm{\infty }:\begin{array}{l}{T}_{{\beta }_1}^{n}x<{\beta }_1^{-nv}\\ T_{{\beta }_2}^{n}y<{\beta }_2^{-nv}\end{array}\text{ for infinitely many }n\in \mathbb{N}\}\text{, }$$ and the uniform approximation exponent $${\widehat{v}}_{{\beta }_1,{\beta }_2}(x,y)=\sup \{0\le \widehat{v}<\mathrm{\infty }:\mathrm{\forall }N\gg 1,\mathrm{\exists }1\le n\le N\text{ such that }\begin{array}{l}{T}_{{\beta }_1}^{n}x<{\beta }_1^{-N\widehat{v}}\\ T_{{\beta }_2}^{n}y<{\beta }_2^{-N\widehat{v}}\end{array}\}\mathrm{.}$$ We calculate the Hausdorff dimension of the intersection $$\{(x,y)\in [0,1{]}^2:{\widehat{v}}_{{\beta }_1,{\beta }_2}(x,y)=\widehat{v}\text{ and }{v}_{{\beta }_1,{\beta }_2}(x,y)=v\}$$ for any $\widehat{v}$ and $v$ satisfying ${\log }_{{\beta }_2}{\beta }_1>\frac{\widehat{v}}{v}(1+v)$. As a corollary, we establish a definite formula for the Hausdorff dimension of the level set of the uniform approximation exponent.