Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay

Khintchine’s theorem on the measure dichotomy for the set of $\psi$-approximable numbers has been generalized to inhomogeneous and higher-dimensional settings. Allen and Ramírez conjectured that the monotonicity condition can be removed in the inhomogeneous $nm=2$ cases. In this paper, we resolve the $(n,m)=(1,2)$ case for $\psi$ satisfying a polynomial decay condition $\psi (q)=O(q^{-\delta })$ for some $\delta >0.$