Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay

Abstract

Khintchine's theorem on the measure dichotomy for the set of $\ddot{I}ˆ$-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 $\ddot{I}ˆ$ satisfying a polynomial decay condition $\ddot{I}ˆ(q)=O(q^{-\hat{I}´})$ for some $\hat{I}´>0.$

7 pages