HAUSDORFF DIMENSION OF THE SET APPROXIMATED BY IRRATIONAL ROTATIONS. Issue 1 (14th February 2018)

• Record Type: Journal Article
• Authors: Kim, Dong Han; Rams, Michał; Wang, Baowei
• Is Part Of: Mathematika. Volume 64:Issue 1(2018)
• Page Start: 267
• Page End: 283
• Publication Date: 2018-02-14
• Subjects: 11K55 (primary) -- 11J71 -- 28A80 (secondary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579317000523
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract : Let θ be an irrational number and φ : N → R + be a monotone decreasing function tending to zero. Let E φ ( θ ) = { y ∈ R : ∥ n θ − y ∥ < φ ( n ), for infinitely many n ∈ N }, i.e. the set of points which are approximated by the irrational rotation with respect to the error function φ ( n ) . In this article, we give a complete description of the Hausdorff dimension of E φ ( θ ) for any monotone function φ and any irrational θ .