Abstract

In their recent Invent. Math. paper, McKinnon and Roth introduced the approximation constant x(L) to an algebraic point x on an algebraic variety V with an ample line bundle bundle L. The invariant x(L) measures how well x can be approximated by rational points on X with respect to the height function associate to L. They showed that x(L) is closely related to the Seshadri constant x(L) measuring the local positivity of L at x. They also showed that the invariant x(L) can be computed through another invariant x(L) in the height inequality they established. By computing the Seshadri constant x(L) for the case of V = P1, their result recovers the Roth's theorem, so the height inequality they established can be viewed as the generalization of the Roth's theorem to arbitrary projective varieties.

In my recent joint work with Min Ru, we give such results a short and simpler proof. Furthermore, we extend the results from points of a projective variety to subschemes. The generalized result in terms of subschemes connects, as well as gives a clearer explanation, the above mentioned result of McKinnon and Roth with the recent Diophantine approximation results in term of the divisors obtained by Autissier, Corvarja, Evertse, Ferretti, Levin, Ru, Vojta, Zannier, ....

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