The current topic, starting from Furstenberg’s conjecture, is to investigate the distribution of orbits under two incommensurable dynamics. In this talk, we consider the Diophantine property about the joint distribution of points under the dynamics of rotation and dyadic expansion. The main task is to determine the Lebesgue measure and Hausdorff dimension of the set $${x: \|nx\|<\varphi(n), \|2^nx\|<\psi(n), i.m. n\in \mathbb{N}}. $$