Dirichlet Uniformly Well-approximated Numbers. (14th February 2018)

• Record Type: Journal Article
• Authors: Kim, Dong Han; Liao, Lingmin
• Is Part Of: International mathematics research notices. Volume 2019:Number 24(2019)
• Page Start: 7691
• Page End: 7732
• Publication Date: 2018-02-14
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rny015
• Languages: English
• ISSNs: 1073-7928

Abstract

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.