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Let $f, g$ be two positive continuous functions. For any $x_0,y_0\\in[0,1]$, define the shrinking target set\n  $$ E(T_\\beta, f,g):=\\left\\{(x,y)\\in [0,1]^2: \\begin{array}{ll} |T_{\\beta}^{n}x-x_{0}|<e^{-S_nf(x)}\\\\ [1ex] |T_{\\beta}^{n}y-y_{0}|< e^{-S_ng(y)} \\end{array} \\ {\\text{for infinitely many}} \\ n\\in \\N \\right\\}, $$ where $S_nf(x)=\\sum_{j=0}^{n-1}f(T_\\beta^jx)$ is the Birkhoff sum. 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