Diophantine Approximation for Subvarieties and GCD problems

Tzu-Yueh Wang

Institute of Mathematics

Academia Sinica

jwang@math.sinica.edu.tw

    A fundamental problem in Diophantine approximation is to study how well an algebraic number can be approximated by rational numbers. The celebrated Roth's theorem states that for a fixed algebraic number `\alpha`, `\epsilon>0`, and `C>0`, there are only finitely many `p/q\in\mathbb{Q}` , where `p` and `q` are relatively prime integers, such that `|\alpha-\frac{p}{q}|\leq\frac{C}{|b|^{2+\epsilon}}`.
There are two kinds of generalization of Roth's theorem. One is to approximate an algebraic point by rational points in an arbitrary projective varieties which is done in a recent work of McKinnon and Roth. Another is in the direction of Schmidt's subspace theorem to study Diophantine approximation of rational points to a set of hyperplanes in projective spaces, or more generally a set of divisors in an arbitrary projective variety.
I will discuss a Diophantine inequality in terms of subschemes which is a joint work with Min Ru. In the case of points, it recovers a result of McKinnon and Roth. In the case of divisors, it connects Schmidt's subspace theorem and the recent Diophantine approximation results obtained by Autissier, Corvarja, Evertse, Faltings, Ferretti, Levin, Ru, W ustholz, Vojta, Zannier, and etc. I will then discuss possible application of the above result to the study of gcd problem which is a joint work with Ji Gou.