Dirichlet Uniformly Well-approximated Numbers. (14th February 2018) - Kim, Dong Han; Liao, Lingmin
Abstract: Fix an irrational number θ . For a real number τ > 0, consider the numbers y satisfying that for all large number Q, there exists an integer 1 ≤ n ≤ Q, such that ∥ nθ − y ∥ < Q − τ, where ∥⋅∥ is the distance of a real number to its nearest integer. These numbers are called Dirichlet uniformly well-approximated numbers. For any τ > 0, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of θ . It is also proved that with respect to τ, the only possible discontinuous point of the Hausdorff dimension is τ = 1.