We outline a proof of an analogue of Khintchine's Theorem in R^2, where the ordinary height is replaced by a distance function satisfying an irrationality condition as well as certain decay and symmetry conditions.
A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard
Variants of Khintchine's theorem in metric Diophantine approximation
Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target
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