A note on badly approximabe sets in projective space

Harrap, S.; Hussain, M.

Abstract

Recently, Ghosh and Haynes (J Reine Angew Math 712:39–50, 2016) proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarník-type result also holds for ‘badly approximable’ points in real projective space. In particular, we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full Hausdorff dimension when intersected with certain compact subsets of real projective space. Furthermore, we also establish an analogue of Khintchine’s theorem for convergence relating to ‘intrinsic’ approximation of points in these compact sets.

Citation

Harrap, S., & Hussain, M. (2017). A note on badly approximabe sets in projective space. Mathematische Zeitschrift, 285(1), 239-250. https://doi.org/10.1007/s00209-016-1705-y

• Journal Article Type: Article
• Acceptance Date: Mar 17, 2016
• Online Publication Date: May 30, 2016
• Publication Date: 2017-02
• Deposit Date: May 15, 2017
• Journal: Mathematische Zeitschrift
• Print ISSN: 0025-5874
• Electronic ISSN: 1432-1823
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 285
• Issue: 1
• Pages: 239-250
• DOI: https://doi.org/10.1007/s00209-016-1705-y