A characterization of bad approximability*For Joaquín Alberto. (2nd August 2018)

• Record Type: Journal Article
• Authors: Ramírez, Felipe A
• Is Part Of: Nonlinearity. Volume 31:Number 9(2018:Sep.)
• Page Start: 4246
• Page End: 4262
• Publication Date: 2018-08-02
• Subjects: Diophantine approximation -- badly approximable vectors -- Kurzweil's theorem -- inhomogeneous approximation -- shrinking target; 11K60 -- 11J20 -- 11J83 -- 37E10; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aaca8e
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We show that badly approximable vectors are exactly those that cannot, for any inhomogeneous parameter, be inhomogeneously approximated at every monotone divergent rate. This implies in particular that Kurzweil's theorem cannot be restricted to any points in the inhomogeneous part. Our results generalize to weighted approximations, and to higher irrationality exponents.