A characterization of bad approximability

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.

19 pages; v2: minor changes, not of a mathematical nature; v3: minor changes, not mathematical

References in corpus (4)

• Badly approximable points on manifolds
• Badly approximable points on planar curves and winning
• Diophantine approximation on planar curves and the distribution of rational points
• An analogue of a theorem of Kurzweil