Xintian Zhang

Hausdorff dimension of the set of \psi-badly approximable points on R^d

The research about how well a real number can be approximated by rationals is an active topic in number theory. The origin of this topic is to find the size of the set of \psi-well approximable numbers. On R^d, an explicit general formula of its Hausdorff dimension is given by Jarnik. The research was continued to find the size of the set of \psi -badly approximable numbers and the size of the set of \psi-exactly approximable numbers. This problem is well understood on R^1. However, the method involved on R^1 relies heavily on continued fractions which do not generalize to R^n. In the joint work with Henna Koivusalo, Benjamin Ward and Jason Levesley, we developed a method based on a modified mass transference principle that allow us to find the Hausdorff dimension of the \psi-badly approximable numbers on R^d.