Seonhee Lim

Effective Hausdorff dimension of Bad sets

In this talk, we consider the inhomogeneous Diophantine approximation: the distribution of q a modulo integers near a target real b (for integer q and a real a), or more generally Aq modulo integral vectors near a target vector b (where q is an integer vector, and A is a real matrix). We prove that for all b, the Hausdorff dimension of the set of matrices that are epsilon badly approximable for the target b is not full, with an effective upper bound. We also give an effective bound on the dimension of the set of targets badly approximated by Aq in terms of epsilon, if the matrix A is not singular on average. The main part of the talk is joint work with Taehyeong Kim and Wooyeon Kim.