Inhomogeneous Diophantine approximation on planar curves

In this paper we develop the inhomogeneous metric theory of simultaneous Diophantine approximation on planar curves. Our results naturally extend the homogeneous Khintchine and Jarník type theorems established in Beresnevich et al. (Ann Math 166(2):367–426, 2007) and Vaughan and Velani (Invent Math 166:103–124, 2006) and are the first of their kind. The key lies in obtaining essentially the best possible results regarding the distribution of ‘shifted’ rational points near planar curves.