Badly approximable points on planar curves and a problem of Davenport

Badziahin, D.; Velani, S.

Abstract

Let C be two times continuously differentiable curve in R2 with at least one point at which the curvature is non-zero. For any i,j⩾0 with i+j=1, let Bad(i,j) denote the set of points (x,y)∈R2 for which max{∥qx∥1/i,∥qy∥1/j}>c/q for all q∈N. Here c=c(x,y) is a positive constant. Our main result implies that any finite intersection of such sets with C has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.

Citation

Badziahin, D., & Velani, S. (2014). Badly approximable points on planar curves and a problem of Davenport. Mathematische Annalen, 359(3-4), 969-1023. https://doi.org/10.1007/s00208-014-1020-z

• Journal Article Type: Article
• Acceptance Date: Oct 15, 2013
• Online Publication Date: Mar 1, 2014
• Publication Date: Aug 1, 2014
• Deposit Date: Nov 26, 2014
• Publicly Available Date: Nov 27, 2014
• Journal: Mathematische Annalen
• Print ISSN: 0025-5831
• Electronic ISSN: 1432-1807
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 359
• Issue: 3-4
• Pages: 969-1023
• DOI: https://doi.org/10.1007/s00208-014-1020-z
• Keywords: 11J83, 11J13, 11K60.
• Public URL: https://durham-repository.worktribe.com/output/1416684