The Cartesian product of shrinking target sets in dyadic system and triadic system

Abstract

In this paper, we consider the Cartesian product of shrinking target sets. Let f and g be two positive continuous functions. For any $x_0,y_0\in [0,1]$, we define the shrinking target sets as follows:$E_2\left(f\right)=\{x\in [0,1]:|T_2^{n}x-x_0|<e^{-S_{n}f\left(x\right)}\text{ for infinitely many }n\in N\},$ and$E_3\left(g\right)=\{y\in [0,1]:|T_3^{m}y-y_0|<e^{-{\overline{S}}_{m}g\left(y\right)}\text{ for infinitely many }m\in N\},$ where $S_{n}f\left(x\right)={\sum }_{j=0}^{n-1}f\left(T_2^{j}x\right)$ and ${\overline{S}}_{m}g\left(y\right)={\sum }_{j=0}^{m-1}g\left(T_3^{j}y\right)$ denote the Birkhoff ergodic sums, and $T_{b}x=bx\left(\text{mod }1\right)$.

The Hausdorff dimension of the Cartesian product set $E_2\left(f\right)\times E_3\left(g\right)$ is determined in this work.