For $\alpha, \beta>3,$ define $$\{(x,y)\in[0,1]\times [0,1]: \|qx\|\le q^{1-\alpha}, \|qy\|\le q^{1-\beta} \quad \text{for infinitely many $ q\in \mathbb{N}$}\}.$$ This set can be regarded as a two dimensional variation of the Jarnik set.
Does anybody know some results on the Hausdorff dimension of this set? Many thanks!
