Schmidt's subspace theorem with moving targets

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In recent years, due to work of Vojta, Lang, Osgood, etc., people have started to realize that there is a close relationship between Nevanlinna theory and Diophantine approximation. P. Vojta has compiled a dictionary (see Vojta [V1]) providing correspondences for the concepts fundamental to the two theories. The second main theorem for meromorphic functions is the most important result in Nevanlinna theory, and its counterpart in number theory, via Vojta’s dictionary, is Roth’s theorem. In 1929, R. Nevanlinna conjectured that the second main theorem remains correct if distinct points in P1(C) are replaced by distinct holomorphic maps ai : C → P1(C) with Tai (r) = o(Tf (r)), where Tf (r) is the Nevanlinna characteristic function of the holomorphic map f : C → P1(C) (see [RS 1] for the definition). After many attempts by various people, Osgood proved the conjecture in 1985, which is now known as the Second Main Theorem with Moving Targets. In 1986, Steinmetz gave another simple and elegant proof. Recently, Vojta ([V 2] and [V 3]) formulated and proved its counterpart in number theory: Theorem A (Roth’s theorem with moving targets). Let k be a number field and let S be a finite set of places of k . Let q ∈ Z>0, and let > 0. Then there does not exist an infinite sequence of distinct tuples