Alladi Ramakrishnan Hall

Approximation of rational points and a characterization of projective spaces

Akash Sengupta

Rutgers University

Given a real number x, how well can we approximate it using rational numbers? This question has been classically studied by Dirichlet, Liouville, Roth et al, and the approximation exponent of a real number x measures how well we can approximate x. Similarly, given an algebraic variety X over a number field k and a point x in X, we can ask how well can we approximate x using k-rational points? McKinnon and Roth generalized the approximation exponent to this setting and showed that several classical results also generalize to rational points algebraic varieties. In this talk, we will define a new variant of the approximation constant which also captures the geometric properties of the variety X. We will see that this geometric approximation constant is closely related to the behaviour of rational curves on X. In particular, I’ll talk about a result showing that if the geometric approximation constant is larger than the dimension of X, then X must be isomorphic to a projective space. This talk is based on joint work with David McKinnon.