This article gives an introductory survey of recent progress on Diophantine problems, especially consequences coming from Schmidt’s subspace theorem, Baker’s transcendence method and Padé approximation. We present fundamental properties around Diophantine approximation and how it yields results in number theory.
In this paper, we establish a Schmidt’s subspace theorem for non-subdegenerate families of hyperplanes. In particular, our result improves the previous result on Schmidt’s subspace type theorem for the case of non-degenerate families of hyperplanes, and furthermore, also shows the sharpness of the condition of non-subdegeneracy. As a consequence, we deduce a version of Lang’s conjecture on exceptional sets in the case of complements of hyperplanes.
In M. Ru and P. Vojta, A birational Nevanlinna constant and its consequences, Amer. J. Math. 142(3) (2020) 957–991, among other things, proved the so-called general theorem (arithmetic part) which can be viewed as an extension of Schmidt’s subspace theorem. In this paper, we extend their result by replacing the divisors by closed subschemes.
A generalized subspace theorem for closed subschemes in subgeneral position
