Badly approximable points on planar curves and a problem of Davenport

Badziahin, D.; Velani, S.

doi:10.1007/s00208-014-1020-z

Badly approximable points on planar curves and a problem of Davenport

Badziahin, D.; Velani, S.

Authors

D. Badziahin

S. Velani

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.