Dirichlet Uniformly Well-approximated Numbers

Abstract

Fix an irrational number.. For a real number tau > 0, consider the numbers y satisfying that for all large number Q, there exists an integer 1 <= n <= Q, such that parallel to n theta - y parallel to < Q(-tau), where parallel to center dot parallel to is the distance of a real number to its nearest integer. These numbers are called Dirichlet uniformly well-approximated numbers. For any tau > 0, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of theta. It is also proved that with respect to tau, the only possible discontinuous point of the Hausdorff dimension is tau = 1.